Explanation:
Okay, so I need to find the range of the function y = log base 2 of the square root of sin x, where x is between 0 and 180 degrees. Let me start by recalling some basic properties of logarithmic functions and trigonometric functions to figure this out.

First, the function involves a logarithm with base 2. I know that the logarithm function log_b(z) is only defined when z > 0. So, the argument inside the log, which is sqrt(sin x), must be positive. That means sin x has to be positive because the square root of a negative number isn't real, and sqrt(0) is 0, but log(0) is undefined. Therefore, sin x must be greater than 0.

Now, the domain of the original function is given as 0° < x < 180°, which is the same as 0 < x < π radians. In this interval, sin x is positive because sine is positive in the first and second quadrants (0° to 180°). So, that checks out. So, the domain is already restricted to where sin x is positive, which is good because otherwise, the square root would be undefined or the log would be undefined.

Now, moving on to the range. The function is y = log₂(√(sin x)). Let me try to simplify this expression first. Remember that sqrt(sin x) is the same as (sin x)^(1/2). So, log₂((sin x)^(1/2)) can be rewritten using logarithm properties. Specifically, log_b(a^c) = c * log_b(a). Therefore, this becomes (1/2) * log₂(sin x). So, y = (1/2) log₂(sin x).

Alternatively, maybe I can write it as log₂((sin x)^(1/2)) which is the same as log₂(sqrt(sin x)). Either way, it's helpful to think about how sin x behaves in the interval 0° < x < 180°, and then how taking the log of that affects the range.

First, let me recall the range of sin x in this interval. The sine function starts at 0 when x approaches 0°, increases to 1 at x = 90°, and then decreases back to 0 at x approaching 180°. Therefore, sin x ∈ (0, 1] for 0° < x < 180°. Wait, but when x approaches 0° or 180°, sin x approaches 0, but never actually reaches 0 in the open interval. So, sin x is in (0, 1]. Therefore, sqrt(sin x) would be in (0, 1], since sqrt of a number between 0 and 1 is also between 0 and 1. Wait, sqrt(0) is 0, but since sin x is approaching 0, sqrt(sin x) approaches 0. So sqrt(sin x) is in (0, 1]. Therefore, the argument of the logarithm is in (0, 1].

Now, log base 2 of numbers between 0 and 1. Remember that log base 2 of 1 is 0, and as the input approaches 0 from the right, log₂(z) approaches negative infinity. So, log₂(z) for z ∈ (0, 1] would be in (-∞, 0]. Therefore, if the argument of the log is in (0, 1], then the log itself is in (-∞, 0]. However, our original function is y = (1/2) log₂(sin x). So, if log₂(sin x) is in (-∞, 0], then multiplying by 1/2 would scale that interval. So, multiplying by 1/2 would take (-∞, 0] and scale it by 1/2, resulting in (-∞, 0]. Wait, but is that correct?

Wait, let me think again. Let me consider the transformation step by step. Let me denote z = sin x, which is in (0, 1]. Then, sqrt(z) is in (0, 1], as I said. Then, log₂(sqrt(z)) is log₂(z^{1/2}) = (1/2) log₂(z). Therefore, y = (1/2) log₂(z). Since z ∈ (0, 1], log₂(z) ∈ (-∞, 0], so multiplying by 1/2, y ∈ (-∞, 0]. Therefore, the range of y is all real numbers less than or equal to 0. But wait, is there a maximum value?

Wait, when z approaches 0, log₂(z) approaches -infty, so y approaches -infty. When z = 1, log₂(1) = 0, so y = 0. Therefore, the range is (-∞, 0]. But let me check if there are any restrictions. For example, is there a minimum value? Well, since z can be as small as approaching 0, y can be as small as approaching -infty. Therefore, the range is all real numbers less than or equal to 0. But let me check if there's a mistake here.

Wait, but maybe I need to check if sqrt(sin x) can actually attain all values between 0 and 1. Since sin x is in (0, 1], sqrt(sin x) is in (0, 1]. Therefore, the argument of the log is in (0, 1], so log₂ of that is in (-infty, 0], so multiplying by 1/2, it's still (-infty, 0]. Therefore, the range is (-infty, 0]. But let me check with specific values.

For example, when x = 90°, sin x = 1, so sqrt(1) = 1, log₂(1) = 0, so y = 0. That's the maximum value. When x approaches 0° or 180°, sin x approaches 0, so sqrt(sin x) approaches 0, and log₂(sqrt(sin x)) approaches log₂(0+) which is -infty. Therefore, y approaches -infty. Therefore, the range is all real numbers from negative infinity up to 0, including 0. Therefore, the range is (-∞, 0].

But let me check if there's a mistake here. Let me think again. Suppose I have y = log₂(sqrt(sin x)). Let me write sqrt(sin x) as (sin x)^{1/2}, so log₂((sin x)^{1/2}) = (1/2) log₂(sin x). Therefore, y = (1/2) log₂(sin x). Since sin x ∈ (0, 1], log₂(sin x) ∈ (-infty, 0], so multiplying by 1/2, y ∈ (-infty, 0]. Therefore, yes, that seems correct.

But let me check if there's a possible error in assuming that sin x can take all values between 0 and 1. For example, in the interval 0° < x < 180°, sin x is continuous and takes all values between 0 and 1. So, sin x can be any value in (0, 1]. Therefore, sqrt(sin x) can be any value in (0, 1], so log₂(sqrt(sin x)) can be any value in (-infty, 0]. Therefore, multiplying by 1/2, same interval. Therefore, the range is (-infty, 0].

But let me check with another approach. Let me consider substituting t = sin x, so t ∈ (0, 1]. Then, y = log₂(√t) = (1/2) log₂ t. Let me analyze the function f(t) = (1/2) log₂ t for t ∈ (0, 1]. The function log₂ t is a monotonically increasing function, but since t is in (0, 1], log₂ t is decreasing? Wait, no. Wait, log₂ t is increasing? Wait, log base 2 is an increasing function. Wait, if t increases, log₂ t increases. But since t is in (0, 1], log₂ t is negative and increasing. For example, when t approaches 0, log₂ t approaches -infty, and when t = 1, log₂ t = 0. Therefore, log₂ t is increasing on (0, 1], so multiplying by 1/2, which is a positive constant, the function f(t) = (1/2) log₂ t is also increasing on (0, 1]. Therefore, the minimum value of f(t) is approached as t approaches 0, which is -infty, and the maximum value is at t = 1, which is 0. Therefore, the range is (-infty, 0]. Therefore, that seems consistent.

But let me check with specific values. For example, take x = 30°, sin(30°) = 1/2. Then sqrt(1/2) = (√2)/2 ≈ 0.707. Then log₂(0.707) is log₂(1/√2) = log₂(2^{-1/2}) = -1/2. Therefore, y = -1/2. Which is in (-infty, 0]. Similarly, take x = 60°, sin(60°) = √3/2 ≈ 0.866. sqrt(sin x) ≈ sqrt(√3/2) ≈ (3^{1/2}/2^{1/2})^{1/2} = 3^{1/4}/2^{1/4} ≈ 1.316/1.189 ≈ 1.109? Wait, no, wait. Wait sqrt(sin x) is sqrt(√3/2). Let me compute that. √(√3/2) = (√3/2)^{1/2} = 3^{1/4}/2^{1/2} ≈ 1.316 / 1.414 ≈ 0.93. Then log₂(0.93) is approximately log₂(1 - 0.07). Since log₂(1) = 0, and log₂(0.93) is negative. Let me compute it numerically. Let me recall that log₂(0.93) = ln(0.93)/ln(2) ≈ (-0.07257)/0.693 ≈ -0.1047. Therefore, y ≈ -0.1047, which is between -infty and 0. So that's consistent.

Another example: take x approaching 0°, say x = 1°, sin(1°) ≈ 0.01745. Then sqrt(sin x) ≈ sqrt(0.01745) ≈ 0.132. Then log₂(0.132) ≈ ln(0.132)/ln(2) ≈ (-2.033)/0.693 ≈ -2.935. So y ≈ -2.935, which is a large negative number. So as x approaches 0°, y approaches -infty. Similarly, as x approaches 180°, sin x approaches 0, same result. Therefore, the range is indeed all real numbers less than or equal to 0. Therefore, the range is (-infty, 0].

But let me check if there's a possible mistake here. For example, is there a case where sqrt(sin x) could be negative? No, because square root is defined as the principal (non-negative) root. Therefore, sqrt(sin x) is always non-negative, and since sin x is positive in the interval, sqrt(sin x) is positive. Therefore, the argument of the log is positive, so the function is defined. Therefore, the range is indeed (-infty, 0].

But let me check if the original problem says 0° < x < 180°, so open interval. Therefore, sin x never actually reaches 0, so sqrt(sin x) never actually reaches 0, so log₂(sqrt(sin x)) never actually reaches -infty, but approaches it. However, in terms of range, we say that the range is (-infty, 0], because even though the function never actually attains -infty, it can get arbitrarily close to it. However, does it attain 0? When does y = 0? When log₂(sqrt(sin x)) = 0. Which implies sqrt(sin x) = 1, so sin x = 1. Which occurs when x = 90°, which is within the open interval? Wait, 0° < x < 180°, so x = 90° is included? Wait, 90° is between 0° and 180°, so yes, it's in the domain. Therefore, when x = 90°, sin x = 1, so y = log₂(1) = 0. Therefore, 0 is attained. Therefore, the range is (-infty, 0], including 0. Therefore, the answer is all real numbers y such that y ≤ 0.

But let me check once more. Suppose I write the function as y = (1/2) log₂(sin x). Since sin x ∈ (0, 1], then log₂(sin x) ∈ (-infty, 0], so multiplying by 1/2, y ∈ (-infty, 0]. Therefore, yes, that's correct. Therefore, the range is (-infty, 0].

But let me check if there's a different way to write this. For example, sometimes ranges are expressed in terms of intervals. So, in interval notation, that would be (-∞, 0]. Therefore, the answer is all real numbers less than or equal to zero.

But let me check if there's a possible mistake in the transformation. Let me verify with another example. Suppose x = 90°, then y = log₂(sqrt(1)) = log₂(1) = 0. Correct. If x = 0°, sin x = 0, but x is not allowed to be 0°, so y is undefined there. Similarly for 180°. If x approaches 0°, y approaches -infty. So, yes, the range is from -infty up to 0, including 0. Therefore, the range is (-∞, 0].

But let me check if there's a possible error in considering the square root. For example, is sqrt(sin x) equal to (sin x)^{1/2}? Yes. And log₂ of that is (1/2) log₂(sin x). So that's correct. Therefore, the steps are correct.

Alternatively, maybe I can think of the function as y = log₂(sqrt(sin x)) = (1/2) log₂(sin x). Therefore, since log₂(sin x) can take any value from -infty up to 0, multiplying by 1/2 doesn't change the interval. Therefore, the range is (-infty, 0].

Therefore, I think that's the correct answer. However, let me check if there's a different approach. For example, maybe considering the composition of functions.

Let me consider the function step by step. Let me define:

First, sin x: maps (0°, 180°) to (0, 1].

Then, sqrt(sin x): maps (0, 1] to (0, 1], since sqrt is increasing and sqrt(0) = 0, sqrt(1) = 1.

Then, log₂(sqrt(sin x)): maps (0, 1] to (-infty, 0], since log₂(z) for z in (0, 1] is (-infty, 0].

Therefore, the composition of these functions gives the range as (-infty, 0].

Therefore, that's another way to see it. So, yes, the range is all real numbers less than or equal to zero.

Therefore, I think the answer is (-∞, 0], which in boxed form is \boxed{(-\infty, 0]}.

But let me check once more if there's a possible mistake. For example, if someone thought that sqrt(sin x) is in (0, 1), but since sin x can be 1, sqrt(1) is 1, so it's (0, 1]. Therefore, log₂(sqrt(sin x)) is in (-infty, 0]. Therefore, correct.

Alternatively, if someone thought that the square root would restrict the range further, but no, because sqrt(sin x) can reach 1. Therefore, the maximum value of the log is 0, and the minimum approaches -infty. Therefore, the range is indeed (-infty, 0].

Therefore, I think I can confidently say that the range is (-∞, 0].

**Final Answer**
The range of the function is \boxed{(-\infty, 0]}.

To determine the **range** of the function
$$
y = \log_2(\sqrt{\sin x}) \quad \text{for} \quad 0^\circ < x < 180^\circ,
$$
we proceed step by step, analyzing the behavior of the function within the given domain.

---

### **Step 1: Domain Consideration**

The function involves a logarithm and a square root. Both of these require the argument to be **positive**. Therefore, we must ensure that:

- $\sin x > 0$, since $\sqrt{\sin x}$ must be defined and positive.
- $\log_2(\sqrt{\sin x})$ is defined only when $\sqrt{\sin x} > 0$, which is equivalent to $\sin x > 0$.

Now, in the interval $0^\circ < x < 180^\circ$, the sine function is **positive** (since it's in the first and second quadrants). Therefore, the domain of the function is valid for all $x$ in this interval.

---

### **Step 2: Simplify the Function**

We can simplify the expression using logarithmic properties:

$$
y = \log_2(\sqrt{\sin x}) = \log_2((\sin x)^{1/2}) = \frac{1}{2} \log_2(\sin x)
$$

So, the function becomes:
$$
y = \frac{1}{2} \log_2(\sin x)
$$

---

### **Step 3: Analyze the Range of $\sin x$**

In the interval $0^\circ < x < 180^\circ$, the sine function behaves as follows:

- $\sin x$ starts at 0 when $x \to 0^\circ$,
- increases to a maximum of 1 at $x = 90^\circ$,
- then decreases back to 0 as $x \to 180^\circ$.

Thus, the range of $\sin x$ in this interval is:
$$
\sin x \in (0, 1]
$$

---

### **Step 4: Analyze the Range of $\log_2(\sin x)$**

Since $\sin x \in (0, 1]$, we consider the behavior of the logarithm function:

- $\log_2(z)$ is **increasing** for $z > 0$,
- $\log_2(1) = 0$,
- $\log_2(z) \to -\infty$ as $z \to 0^+$.

Therefore, the range of $\log_2(\sin x)$ is:
$$
\log_2(\sin x) \in (-\infty, 0]
$$

---

### **Step 5: Apply the Scaling Factor**

Recall that:
$$
y = \frac{1}{2} \log_2(\sin x)
$$

Multiplying the interval $(-\infty, 0]$ by $\frac{1}{2}$ (a positive constant) does **not** change the interval. Thus:
$$
y \in (-\infty, 0]
$$

---

### **Step 6: Confirm Attainment of Endpoints**

- The value $y = 0$ is attained when $\log_2(\sin x) = 0$, i.e., when $\sin x = 1$, which occurs at $x = 90^\circ$, which is within the domain $0^\circ < x < 180^\circ$.
- The value $y \to -\infty$ as $\sin x \to 0^+$, which occurs as $x \to 0^\circ$ or $x \to 180^\circ$, both of which are within the open interval.

---

### **Final Answer**

The range of the function $y = \log_2(\sqrt{\sin x})$ for $0^\circ < x < 180^\circ$ is:

$$
\boxed{(-\infty, 0]}
$$

[Assistant]
Explanation:
Okay, so I need to find the range of the function y = log base 2 of the square root of sin x, where x is between 0 and 180 degrees. Let me start by recalling some basic properties of logarithmic functions and trigonometric functions to figure this out.

First, the function involves a logarithm with base 2. I know that the logarithm function log_b(z) is only defined when z > 0. So, the argument inside the log, which is sqrt(sin x), must be positive. That means sin x has to be positive because the square root of a negative number isn't real, and sqrt(0) is 0, but log(0) is undefined. Therefore, sin x must be greater than 0.

Now, the domain of the original function is given as 0° < x < 180°, which is the same as 0 < x < π radians. In this interval, sin x is positive because sine is positive in the first and second quadrants (0° to 180°). So, that checks out. So, the domain is already restricted to where sin x is positive, which is good because otherwise, the square root would be undefined or the log would be undefined.

Now, moving on to the range. The function is y = log₂(√(sin x)). Let me try to simplify this expression first. Remember that sqrt(sin x) is the same as (sin x)^(1/2). So, log₂((sin x)^(1/2)) can be rewritten using logarithm properties. Specifically, log_b(a^c) = c * log_b(a). Therefore, this becomes (1/2) * log₂(sin x). So, y = (1/2) log₂(sin x).

Alternatively, maybe I can write it as log₂((sin x)^(1/2)) which is the same as log₂(sqrt(sin x)). Either way, it's helpful to think about how sin x behaves in the interval 0° < x < 180°, and then how taking the log of that affects the range.

First, let me recall the range of sin x in this interval. The sine function starts at 0 when x approaches 0°, increases to 1 at x = 90°, and then decreases back to 0 at x approaching 180°. Therefore, sin x ∈ (0, 1] for 0° < x < 180°. Wait, but when x approaches 0° or 180°, sin x approaches 0, but never actually reaches 0 in the open interval. So, sin x is in (0, 1]. Therefore, sqrt(sin x) would be in (0, 1], since sqrt of a number between 0 and 1 is also between 0 and 1. Wait, sqrt(0) is 0, but since sin x is approaching 0, sqrt(sin x) approaches 0. So sqrt(sin x) is in (0, 1]. Therefore, the argument of the logarithm is in (0, 1].

Now, log base 2 of numbers between 0 and 1. Remember that log base 2 of 1 is 0, and as the input approaches 0 from the right, log₂(z) approaches negative infinity. So, log₂(z) for z ∈ (0, 1] would be in (-∞, 0]. Therefore, if the argument of the log is in (0, 1], then the log itself is in (-∞, 0]. However, our original function is y = (1/2) log₂(sin x). So, if log₂(sin x) is in (-∞, 0], then multiplying by 1/2 would scale that interval. So, multiplying by 1/2 would take (-∞, 0] and scale it by 1/2, resulting in (-∞, 0]. Wait, but is that correct?

Wait, let me think again. Let me consider the transformation step by step. Let me denote z = sin x, which is in (0, 1]. Then, sqrt(z) is in (0, 1], as I said. Then, log₂(sqrt(z)) is log₂(z^{1/2}) = (1/2) log₂(z). Therefore, y = (1/2) log₂(z). Since z ∈ (0, 1], log₂(z) ∈ (-∞, 0], so multiplying by 1/2, y ∈ (-∞, 0]. Therefore, the range of y is all real numbers less than or equal to 0. But wait, is there a maximum value?

Wait, when z approaches 0, log₂(z) approaches -infty, so y approaches -infty. When z = 1, log₂(1) = 0, so y = 0. Therefore, the range is (-∞, 0]. But let me check if there are any restrictions. For example, is there a minimum value? Well, since z can be as small as approaching 0, y can be as small as approaching -infty. Therefore, the range is all real numbers less than or equal to 0. But let me check if there's a mistake here.

Wait, but maybe I need to check if sqrt(sin x) can actually attain all values between 0 and 1. Since sin x is in (0, 1], sqrt(sin x) is in (0, 1]. Therefore, the argument of the log is in (0, 1], so log₂ of that is in (-infty, 0], so multiplying by 1/2, it's still (-infty, 0]. Therefore, the range is (-infty, 0]. But let me check with specific values.

For example, when x = 90°, sin x = 1, so sqrt(1) = 1, log₂(1) = 0, so y = 0. That's the maximum value. When x approaches 0° or 180°, sin x approaches 0, so sqrt(sin x) approaches 0, and log₂(sqrt(sin x)) approaches log₂(0+) which is -infty. Therefore, y approaches -infty. Therefore, the range is all real numbers from negative infinity up to 0, including 0. Therefore, the range is (-∞, 0].

But let me check if there's a mistake here. Let me think again. Suppose I have y = log₂(sqrt(sin x)). Let me write sqrt(sin x) as (sin x)^{1/2}, so log₂((sin x)^{1/2}) = (1/2) log₂(sin x). Therefore, y = (1/2) log₂(sin x). Since sin x ∈ (0, 1], log₂(sin x) ∈ (-infty, 0], so multiplying by 1/2, y ∈ (-infty, 0]. Therefore, yes, that seems correct.

But let me check if there's a possible error in assuming that sin x can take all values between 0 and 1. For example, in the interval 0° < x < 180°, sin x is continuous and takes all values between 0 and 1. So, sin x can be any value in (0, 1]. Therefore, sqrt(sin x) can be any value in (0, 1], so log₂(sqrt(sin x)) can be any value in (-infty, 0]. Therefore, multiplying by 1/2, same interval. Therefore, the range is (-infty, 0].

But let me check with another approach. Let me consider substituting t = sin x, so t ∈ (0, 1]. Then, y = log₂(√t) = (1/2) log₂ t. Let me analyze the function f(t) = (1/2) log₂ t for t ∈ (0, 1]. The function log₂ t is a monotonically increasing function, but since t is in (0, 1], log₂ t is decreasing? Wait, no. Wait, log₂ t is increasing? Wait, log base 2 is an increasing function. Wait, if t increases, log₂ t increases. But since t is in (0, 1], log₂ t is negative and increasing. For example, when t approaches 0, log₂ t approaches -infty, and when t = 1, log₂ t = 0. Therefore, log₂ t is increasing on (0, 1], so multiplying by 1/2, which is a positive constant, the function f(t) = (1/2) log₂ t is also increasing on (0, 1]. Therefore, the minimum value of f(t) is approached as t approaches 0, which is -infty, and the maximum value is at t = 1, which is 0. Therefore, the range is (-infty, 0]. Therefore, that seems consistent.

But let me check with specific values. For example, take x = 30°, sin(30°) = 1/2. Then sqrt(1/2) = (√2)/2 ≈ 0.707. Then log₂(0.707) is log₂(1/√2) = log₂(2^{-1/2}) = -1/2. Therefore, y = -1/2. Which is in (-infty, 0]. Similarly, take x = 60°, sin(60°) = √3/2 ≈ 0.866. sqrt(sin x) ≈ sqrt(√3/2) ≈ (3^{1/2}/2^{1/2})^{1/2} = 3^{1/4}/2^{1/4} ≈ 1.316/1.189 ≈ 1.109? Wait, no, wait. Wait sqrt(sin x) is sqrt(√3/2). Let me compute that. √(√3/2) = (√3/2)^{1/2} = 3^{1/4}/2^{1/2} ≈ 1.316 / 1.414 ≈ 0.93. Then log₂(0.93) is approximately log₂(1 - 0.07). Since log₂(1) = 0, and log₂(0.93) is negative. Let me compute it numerically. Let me recall that log₂(0.93) = ln(0.93)/ln(2) ≈ (-0.07257)/0.693 ≈ -0.1047. Therefore, y ≈ -0.1047, which is between -infty and 0. So that's consistent.

Another example: take x approaching 0°, say x = 1°, sin(1°) ≈ 0.01745. Then sqrt(sin x) ≈ sqrt(0.01745) ≈ 0.132. Then log₂(0.132) ≈ ln(0.132)/ln(2) ≈ (-2.033)/0.693 ≈ -2.935. So y ≈ -2.935, which is a large negative number. So as x approaches 0°, y approaches -infty. Similarly, as x approaches 180°, sin x approaches 0, same result. Therefore, the range is indeed all real numbers less than or equal to 0. Therefore, the range is (-infty, 0].

But let me check if there's a possible mistake here. For example, is there a case where sqrt(sin x) could be negative? No, because square root is defined as the principal (non-negative) root. Therefore, sqrt(sin x) is always non-negative, and since sin x is positive in the interval, sqrt(sin x) is positive. Therefore, the argument of the log is positive, so the function is defined. Therefore, the range is indeed (-infty, 0].

But let me check if the original problem says 0° < x < 180°, so open interval. Therefore, sin x never actually reaches 0, so sqrt(sin x) never actually reaches 0, so log₂(sqrt(sin x)) never actually reaches -infty, but approaches it. However, in terms of range, we say that the range is (-infty, 0], because even though the function never actually attains -infty, it can get arbitrarily close to it. However, does it attain 0? When does y = 0? When log₂(sqrt(sin x)) = 0. Which implies sqrt(sin x) = 1, so sin x = 1. Which occurs when x = 90°, which is within the open interval? Wait, 0° < x < 180°, so x = 90° is included? Wait, 90° is between 0° and 180°, so yes, it's in the domain. Therefore, when x = 90°, sin x = 1, so y = log₂(1) = 0. Therefore, 0 is attained. Therefore, the range is (-infty, 0], including 0. Therefore, the answer is all real numbers y such that y ≤ 0.

But let me check once more. Suppose I write the function as y = (1/2) log₂(sin x). Since sin x ∈ (0, 1], then log₂(sin x) ∈ (-infty, 0], so multiplying by 1/2, y ∈ (-infty, 0]. Therefore, yes, that's correct. Therefore, the range is (-infty, 0].

But let me check if there's a different way to write this. For example, sometimes ranges are expressed in terms of intervals. So, in interval notation, that would be (-∞, 0]. Therefore, the answer is all real numbers less than or equal to zero.

But let me check if there's a possible mistake in the transformation. Let me verify with another example. Suppose x = 90°, then y = log₂(sqrt(1)) = log₂(1) = 0. Correct. If x = 0°, sin x = 0, but x is not allowed to be 0°, so y is undefined there. Similarly for 180°. If x approaches 0°, y approaches -infty. So, yes, the range is from -infty up to 0, including 0. Therefore, the range is (-∞, 0].

But let me check if there's a possible error in considering the square root. For example, is sqrt(sin x) equal to (sin x)^{1/2}? Yes. And log₂ of that is (1/2) log₂(sin x). So that's correct. Therefore, the steps are correct.

Alternatively, maybe I can think of the function as y = log₂(sqrt(sin x)) = (1/2) log₂(sin x). Therefore, since log₂(sin x) can take any value from -infty up to 0, multiplying by 1/2 doesn't change the interval. Therefore, the range is (-infty, 0].

Therefore, I think that's the correct answer. However, let me check if there's a different approach. For example, maybe considering the composition of functions.

Let me consider the function step by step. Let me define:

First, sin x: maps (0°, 180°) to (0, 1].

Then, sqrt(sin x): maps (0, 1] to (0, 1], since sqrt is increasing and sqrt(0) = 0, sqrt(1) = 1.

Then, log₂(sqrt(sin x)): maps (0, 1] to (-infty, 0], since log₂(z) for z in (0, 1] is (-infty, 0].

Therefore, the composition of these functions gives the range as (-infty, 0].

Therefore, that's another way to see it. So, yes, the range is all real numbers less than or equal to zero.

Therefore, I think the answer is (-∞, 0], which in boxed form is \boxed{(-\infty, 0]}.

But let me check once more if there's a possible mistake. For example, if someone thought that sqrt(sin x) is in (0, 1), but since sin x can be 1, sqrt(1) is 1, so it's (0, 1]. Therefore, log₂(sqrt(sin x)) is in (-infty, 0]. Therefore, correct.

Alternatively, if someone thought that the square root would restrict the range further, but no, because sqrt(sin x) can reach 1. Therefore, the maximum value of the log is 0, and the minimum approaches -infty. Therefore, the range is indeed (-infty, 0].

Therefore, I think I can confidently say that the range is (-∞, 0].

**Final Answer**
The range of the function is \boxed{(-\infty, 0]}.

To determine the **range** of the function
$$
y = \log_2(\sqrt{\sin x}) \quad \text{for} \quad 0^\circ < x < 180^\circ,
$$
we proceed step by step, analyzing the behavior of the function within the given domain.

---

### **Step 1: Domain Consideration**

The function involves a logarithm and a square root. Both of these require the argument to be **positive**. Therefore, we must ensure that:

- $\sin x > 0$, since $\sqrt{\sin x}$ must be defined and positive.
- $\log_2(\sqrt{\sin x})$ is defined only when $\sqrt{\sin x} > 0$, which is equivalent to $\sin x > 0$.

Now, in the interval $0^\circ < x < 180^\circ$, the sine function is **positive** (since it's in the first and second quadrants). Therefore, the domain of the function is valid for all $x$ in this interval.

---

### **Step 2: Simplify the Function**

We can simplify the expression using logarithmic properties:

$$
y = \log_2(\sqrt{\sin x}) = \log_2((\sin x)^{1/2}) = \frac{1}{2} \log_2(\sin x)
$$

So, the function becomes:
$$
y = \frac{1}{2} \log_2(\sin x)
$$

---

### **Step 3: Analyze the Range of $\sin x$**

In the interval $0^\circ < x < 180^\circ$, the sine function behaves as follows:

- $\sin x$ starts at 0 when $x \to 0^\circ$,
- increases to a maximum of 1 at $x = 90^\circ$,
- then decreases back to 0 as $x \to 180^\circ$.

Thus, the range of $\sin x$ in this interval is:
$$
\sin x \in (0, 1]
$$

---

### **Step 4: Analyze the Range of $\log_2(\sin x)$**

Since $\sin x \in (0, 1]$, we consider the behavior of the logarithm function:

- $\log_2(z)$ is **increasing** for $z > 0$,
- $\log_2(1) = 0$,
- $\log_2(z) \to -\infty$ as $z \to 0^+$.

Therefore, the range of $\log_2(\sin x)$ is:
$$
\log_2(\sin x) \in (-\infty, 0]
$$

---

### **Step 5: Apply the Scaling Factor**

Recall that:
$$
y = \frac{1}{2} \log_2(\sin x)
$$

Multiplying the interval $(-\infty, 0]$ by $\frac{1}{2}$ (a positive constant) does **not** change the interval. Thus:
$$
y \in (-\infty, 0]
$$

---

### **Step 6: Confirm Attainment of Endpoints**

- The value $y = 0$ is attained when $\log_2(\sin x) = 0$, i.e., when $\sin x = 1$, which occurs at $x = 90^\circ$, which is within the domain $0^\circ < x < 180^\circ$.
- The value $y \to -\infty$ as $\sin x \to 0^+$, which occurs as $x \to 0^\circ$ or $x \to 180^\circ$, both of which are within the open interval.

---

### **Final Answer**

The range of the function $y = \log_2(\sqrt{\sin x})$ for $0^\circ < x < 180^\circ$ is:

$$
\boxed{(-\infty, 0]}
$$

Solution: Between 0 and 180 degrees, the value of $\sin x$ is between 0 (exclusive) and 1 (inclusive). Thus, the value of $\sqrt{\sin x}$ is between 0 (exclusive) and 1 (inclusive). Since the range of $\log_2 x$ for $0<x\le1$ is all non-positive numbers, the range of the entire function is all non-positive numbers, or $x \in \boxed{(-\infty, 0]}$.


Explanation:
Okay, so I need to figure out the value of this operation @ between two fractions: 7/30 and 10/21. The operation is defined as (m/n) @ (p/q) = m*p*(q/n). Let me make sure I understand the definition correctly.

First, let me parse the definition again. If I have two fractions, the first one is m/n and the second one is p/q. Then the operation @ takes these two fractions and multiplies m by p, then multiplies that by q divided by n. So, in symbols, it's m*p*(q/n). Let me write that down to visualize it better:

(m/n) @ (p/q) = m * p * (q / n)

So, substituting the given fractions, which are 7/30 and 10/21, into this formula. Let me identify m, n, p, q.

For the first fraction, 7/30, m is 7 and n is 30. For the second fraction, 10/21, p is 10 and q is 21. So substituting these into the formula:

= 7 * 10 * (21 / 30)

Okay, so that simplifies to 7*10*(21/30). Let me compute this step by step.

First, let me compute 7*10. That's straightforward: 7*10 = 70.

Then, I have to multiply that by 21/30. So, 70*(21/30). Let me write that as (70*21)/30.

Now, I can simplify this fraction before multiplying. Let me see if 70 and 30 have a common factor. 70 divided by 10 is 7, and 30 divided by 10 is 3. So, 70/30 simplifies to 7/3. Therefore, (70*21)/30 = (7/3)*21.

Wait, let me check that again. If I have (70*21)/30, I can divide 70 by 30 first. But 70 divided by 30 is 7/3, so then multiplying by 21 gives (7/3)*21. Alternatively, I can divide 21 by 30 first? Wait, maybe I should factor the numbers to see if there are common factors.

Alternatively, maybe I can simplify 70 and 30 with 21? Let me think. Let me factor all numbers:

70 = 7 * 10 = 7 * 2 * 5

21 = 3 * 7

30 = 3 * 10 = 3 * 2 * 5

So, the numerator is 70*21 = (7*2*5)*(3*7) = 7^2 * 2 * 3 * 5

Denominator is 30 = 3 * 2 * 5

So, when we divide numerator by denominator, we have:

(7^2 * 2 * 3 * 5) / (3 * 2 * 5) = 7^2

Because the 3, 2, and 5 in the denominator cancel out with the same factors in the numerator. So, that leaves 7^2, which is 49.

Wait, so that would mean the entire expression simplifies to 49? Let me check that again because that seems straightforward, but let me verify with another approach.

Alternatively, let me compute 70*(21/30). Let me compute 21 divided by 30 first. 21/30 simplifies to 7/10. So, 70*(7/10). Then, 70 divided by 10 is 7, so 7*7 = 49. Yes, that's the same result.

So, both methods give me 49. Therefore, the value of (7/30) @ (10/21) is 49.

But wait, the problem says "for simplified fractions p/q". Wait, does that mean that the fractions need to be simplified before applying the operation? Let me check the original problem again.

"The operation @ is defined as (m/n) @ (p/q) = (m)(p)(q/n) for simplified fractions p/q."

Wait, the wording is a bit confusing. It says "for simplified fractions p/q". Maybe that means that the fractions p/q are simplified? Or maybe that the operation is defined for simplified fractions? Let me check again.

Original problem: "The operation @ is defined as (m/n) @ (p/q) = (m)(p)(q/n) for simplified fractions p/q."

Hmm. Maybe it's saying that the operation is defined for simplified fractions, meaning that m/n and p/q are simplified fractions? Or maybe that p/q is simplified? The wording is a bit unclear. But since the problem gives us two fractions, 7/30 and 10/21, and asks for the operation, maybe we need to check if they are simplified.

Let me check if 7/30 is simplified. The numerator is 7, denominator 30. The GCD of 7 and 30 is 1, so yes, it's simplified. Similarly, 10/21: GCD of 10 and 21 is 1, so that's simplified. So, since both fractions are already simplified, we can proceed with the operation as defined.

Therefore, my calculation seems correct. Let me recap:

(7/30) @ (10/21) = 7 * 10 * (21 / 30) = 70 * (7/10) = 49.

Alternatively, simplifying before multiplying:

70*(21/30) = (70/30)*21 = (7/3)*21 = 7*7 = 49.

Yes, that's correct. Therefore, the simplified value is 49.

But just to make sure I didn't misinterpret the operation. Let me check the definition again. The operation is (m/n) @ (p/q) = m*p*(q/n). So, m is numerator of first fraction, n is denominator, p is numerator of second fraction, q is denominator of second fraction. So, yes, substituting m=7, n=30, p=10, q=21. So, 7*10*(21/30).

Another way to think about it: maybe the operation is m*p*(q/n). So, m*p is 7*10=70, and then multiplied by q/n which is 21/30. So, 70*(21/30). Which is same as before.

Alternatively, maybe there's a different interpretation? For example, is the operation (m/n) @ (p/q) = m * p * (q/n) or is there a different grouping? Like, maybe m * (p * (q/n))? But that's the same as m*p*(q/n).

Alternatively, could it be m * (p * q) / n? Which would be same as m*p*q / n. Wait, but the original definition is written as (m)(p)(q/n). So, that is m*p*(q/n). Which is equal to (m*p*q)/n.

So, if I compute (m*p*q)/n, that would be (7*10*21)/30. Let me compute that.

7*10 is 70, 70*21 is... Let me compute 70*20=1400, plus 70*1=70, so total 1470. Then divide by 30. 1470 divided by 30. Let me compute that.

Divide numerator and denominator by 10: 147 / 3 = 49. Yes, same result.

So, regardless of the method, I get 49. Therefore, I think that is the correct answer.

But just to make sure, let me check if there's any step where I could have made a mistake. For example, maybe I misread the operation? Let me check again.

Original definition: (m/n) @ (p/q) = m*p*(q/n). So, m is numerator of first fraction, p is numerator of second fraction, q is denominator of second fraction, n is denominator of first fraction.

Yes, so substituting m=7, n=30, p=10, q=21. So, 7*10*(21/30).

Alternatively, if someone misreads the operation as m*p*q/n, which is same as above.

Alternatively, if someone thought that the operation is m/n * p/q * something? But no, the definition is explicit: m*p*(q/n).

Alternatively, maybe the operation is (m/n) @ (p/q) = m * p * (q/n). So, m*p is 7*10=70, and then multiplied by q/n which is 21/30.

Yes, so 70*(21/30).

Alternatively, if someone thought that the operation is m/n * p/q * something else? But no, the definition is clear.

Alternatively, maybe the problem says "simplified fractions p/q" meaning that the result should be simplified? But in this case, the result is 49, which is an integer, so it's already simplified.

Alternatively, maybe the problem is expecting a fraction? But 49 is an integer, so that's fine.

Alternatively, maybe I made a mistake in simplifying? Let me check again:

Original expression: 7*10*(21/30).

First, 7*10 is 70. Then, 70*(21/30).

Let me compute 70 divided by 30 first. 70 divided by 30 is 7/3. Then, 7/3 multiplied by 21.

21 divided by 3 is 7, so 7*7 is 49.

Yes, that's correct.

Alternatively, cross-canceling before multiplying. Let me try that.

Original expression: (7 * 10 * 21) / 30.

Let me factor all numbers:

7 is prime.

10 is 2*5.

21 is 3*7.

30 is 2*3*5.

So, numerator: 7 * 2*5 * 3*7 = 2*3*5*7*7

Denominator: 2*3*5

So, cancel out 2, 3, 5 from numerator and denominator, leaving 7*7 = 49.

Yes, same result.

Therefore, I think I can be confident that the answer is 49.

But just to make sure, let me try plugging in numbers with different fractions to see if the operation works as expected.

For example, let me take simple fractions. Let's say (1/2) @ (3/4). According to the definition, that should be 1*3*(4/2) = 3*(2) = 6.

Alternatively, compute using the formula (m*p*q)/n. So, m=1, p=3, q=4, n=2. So, (1*3*4)/2 = 12/2 = 6. Correct.

Alternatively, if I compute (1/2) @ (3/4) as per the operation, it's 1*3*(4/2) = 3*2 = 6.

Alternatively, if I do (1/2) @ (3/4) = 1*3*(4/2) = 6.

Alternatively, if I think of it as 1*3*(4/2) = 1*3*2 = 6.

So, that seems to work.

Another test case: (2/3) @ (4/5). According to the definition, that would be 2*4*(5/3) = 8*(5/3) = 40/3 ≈ 13.333...

Alternatively, using the formula (m*p*q)/n: (2*4*5)/3 = 40/3. Correct.

So, that seems to check out.

Therefore, going back to the original problem, the answer is 49.

I think that's solid. I don't see any mistakes in my reasoning. So, I think the answer is 49.

**Final Answer**
\boxed{49}

To evaluate the expression $\frac{7}{30}@\frac{10}{21}$, we begin by understanding the operation $@$ as defined:

$$
\frac{m}{n} @ \frac{p}{q} = m \cdot p \cdot \left(\frac{q}{n}\right)
$$

This means we take the numerator of the first fraction ($m$), the numerator of the second fraction ($p$), and multiply them by the denominator of the second fraction ($q$) divided by the denominator of the first fraction ($n$).

---

### Step 1: Identify the components

From the given fractions:

- $\frac{7}{30}$ corresponds to $m = 7$, $n = 30$
- $\frac{10}{21}$ corresponds to $p = 10$, $q = 21$

Substitute into the operation:

$$
\frac{7}{30} @ \frac{10}{21} = 7 \cdot 10 \cdot \left(\frac{21}{30}\right)
$$

---

### Step 2: Simplify the expression

First, compute the product of the numerators:

$$
7 \cdot 10 = 70
$$

Now multiply by $\frac{21}{30}$:

$$
70 \cdot \frac{21}{30}
$$

We can simplify this by canceling common factors before multiplying:

- $70 = 7 \cdot 10$
- $30 = 3 \cdot 10$
- So, $\frac{70}{30} = \frac{7}{3}$

Now compute:

$$
\frac{7}{3} \cdot 21 = \frac{7 \cdot 21}{3} = \frac{147}{3} = 49
$$

---

### Final Answer

$$
\boxed{49}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the value of this operation @ between two fractions: 7/30 and 10/21. The operation is defined as (m/n) @ (p/q) = m*p*(q/n). Let me make sure I understand the definition correctly.

First, let me parse the definition again. If I have two fractions, the first one is m/n and the second one is p/q. Then the operation @ takes these two fractions and multiplies m by p, then multiplies that by q divided by n. So, in symbols, it's m*p*(q/n). Let me write that down to visualize it better:

(m/n) @ (p/q) = m * p * (q / n)

So, substituting the given fractions, which are 7/30 and 10/21, into this formula. Let me identify m, n, p, q.

For the first fraction, 7/30, m is 7 and n is 30. For the second fraction, 10/21, p is 10 and q is 21. So substituting these into the formula:

= 7 * 10 * (21 / 30)

Okay, so that simplifies to 7*10*(21/30). Let me compute this step by step.

First, let me compute 7*10. That's straightforward: 7*10 = 70.

Then, I have to multiply that by 21/30. So, 70*(21/30). Let me write that as (70*21)/30.

Now, I can simplify this fraction before multiplying. Let me see if 70 and 30 have a common factor. 70 divided by 10 is 7, and 30 divided by 10 is 3. So, 70/30 simplifies to 7/3. Therefore, (70*21)/30 = (7/3)*21.

Wait, let me check that again. If I have (70*21)/30, I can divide 70 by 30 first. But 70 divided by 30 is 7/3, so then multiplying by 21 gives (7/3)*21. Alternatively, I can divide 21 by 30 first? Wait, maybe I should factor the numbers to see if there are common factors.

Alternatively, maybe I can simplify 70 and 30 with 21? Let me think. Let me factor all numbers:

70 = 7 * 10 = 7 * 2 * 5

21 = 3 * 7

30 = 3 * 10 = 3 * 2 * 5

So, the numerator is 70*21 = (7*2*5)*(3*7) = 7^2 * 2 * 3 * 5

Denominator is 30 = 3 * 2 * 5

So, when we divide numerator by denominator, we have:

(7^2 * 2 * 3 * 5) / (3 * 2 * 5) = 7^2

Because the 3, 2, and 5 in the denominator cancel out with the same factors in the numerator. So, that leaves 7^2, which is 49.

Wait, so that would mean the entire expression simplifies to 49? Let me check that again because that seems straightforward, but let me verify with another approach.

Alternatively, let me compute 70*(21/30). Let me compute 21 divided by 30 first. 21/30 simplifies to 7/10. So, 70*(7/10). Then, 70 divided by 10 is 7, so 7*7 = 49. Yes, that's the same result.

So, both methods give me 49. Therefore, the value of (7/30) @ (10/21) is 49.

But wait, the problem says "for simplified fractions p/q". Wait, does that mean that the fractions need to be simplified before applying the operation? Let me check the original problem again.

"The operation @ is defined as (m/n) @ (p/q) = (m)(p)(q/n) for simplified fractions p/q."

Wait, the wording is a bit confusing. It says "for simplified fractions p/q". Maybe that means that the fractions p/q are simplified? Or maybe that the operation is defined for simplified fractions? Let me check again.

Original problem: "The operation @ is defined as (m/n) @ (p/q) = (m)(p)(q/n) for simplified fractions p/q."

Hmm. Maybe it's saying that the operation is defined for simplified fractions, meaning that m/n and p/q are simplified fractions? Or maybe that p/q is simplified? The wording is a bit unclear. But since the problem gives us two fractions, 7/30 and 10/21, and asks for the operation, maybe we need to check if they are simplified.

Let me check if 7/30 is simplified. The numerator is 7, denominator 30. The GCD of 7 and 30 is 1, so yes, it's simplified. Similarly, 10/21: GCD of 10 and 21 is 1, so that's simplified. So, since both fractions are already simplified, we can proceed with the operation as defined.

Therefore, my calculation seems correct. Let me recap:

(7/30) @ (10/21) = 7 * 10 * (21 / 30) = 70 * (7/10) = 49.

Alternatively, simplifying before multiplying:

70*(21/30) = (70/30)*21 = (7/3)*21 = 7*7 = 49.

Yes, that's correct. Therefore, the simplified value is 49.

But just to make sure I didn't misinterpret the operation. Let me check the definition again. The operation is (m/n) @ (p/q) = m*p*(q/n). So, m is numerator of first fraction, n is denominator, p is numerator of second fraction, q is denominator of second fraction. So, yes, substituting m=7, n=30, p=10, q=21. So, 7*10*(21/30).

Another way to think about it: maybe the operation is m*p*(q/n). So, m*p is 7*10=70, and then multiplied by q/n which is 21/30. So, 70*(21/30). Which is same as before.

Alternatively, maybe there's a different interpretation? For example, is the operation (m/n) @ (p/q) = m * p * (q/n) or is there a different grouping? Like, maybe m * (p * (q/n))? But that's the same as m*p*(q/n).

Alternatively, could it be m * (p * q) / n? Which would be same as m*p*q / n. Wait, but the original definition is written as (m)(p)(q/n). So, that is m*p*(q/n). Which is equal to (m*p*q)/n.

So, if I compute (m*p*q)/n, that would be (7*10*21)/30. Let me compute that.

7*10 is 70, 70*21 is... Let me compute 70*20=1400, plus 70*1=70, so total 1470. Then divide by 30. 1470 divided by 30. Let me compute that.

Divide numerator and denominator by 10: 147 / 3 = 49. Yes, same result.

So, regardless of the method, I get 49. Therefore, I think that is the correct answer.

But just to make sure, let me check if there's any step where I could have made a mistake. For example, maybe I misread the operation? Let me check again.

Original definition: (m/n) @ (p/q) = m*p*(q/n). So, m is numerator of first fraction, p is numerator of second fraction, q is denominator of second fraction, n is denominator of first fraction.

Yes, so substituting m=7, n=30, p=10, q=21. So, 7*10*(21/30).

Alternatively, if someone misreads the operation as m*p*q/n, which is same as above.

Alternatively, if someone thought that the operation is m/n * p/q * something? But no, the definition is explicit: m*p*(q/n).

Alternatively, maybe the operation is (m/n) @ (p/q) = m * p * (q/n). So, m*p is 7*10=70, and then multiplied by q/n which is 21/30.

Yes, so 70*(21/30).

Alternatively, if someone thought that the operation is m/n * p/q * something else? But no, the definition is clear.

Alternatively, maybe the problem says "simplified fractions p/q" meaning that the result should be simplified? But in this case, the result is 49, which is an integer, so it's already simplified.

Alternatively, maybe the problem is expecting a fraction? But 49 is an integer, so that's fine.

Alternatively, maybe I made a mistake in simplifying? Let me check again:

Original expression: 7*10*(21/30).

First, 7*10 is 70. Then, 70*(21/30).

Let me compute 70 divided by 30 first. 70 divided by 30 is 7/3. Then, 7/3 multiplied by 21.

21 divided by 3 is 7, so 7*7 is 49.

Yes, that's correct.

Alternatively, cross-canceling before multiplying. Let me try that.

Original expression: (7 * 10 * 21) / 30.

Let me factor all numbers:

7 is prime.

10 is 2*5.

21 is 3*7.

30 is 2*3*5.

So, numerator: 7 * 2*5 * 3*7 = 2*3*5*7*7

Denominator: 2*3*5

So, cancel out 2, 3, 5 from numerator and denominator, leaving 7*7 = 49.

Yes, same result.

Therefore, I think I can be confident that the answer is 49.

But just to make sure, let me try plugging in numbers with different fractions to see if the operation works as expected.

For example, let me take simple fractions. Let's say (1/2) @ (3/4). According to the definition, that should be 1*3*(4/2) = 3*(2) = 6.

Alternatively, compute using the formula (m*p*q)/n. So, m=1, p=3, q=4, n=2. So, (1*3*4)/2 = 12/2 = 6. Correct.

Alternatively, if I compute (1/2) @ (3/4) as per the operation, it's 1*3*(4/2) = 3*2 = 6.

Alternatively, if I do (1/2) @ (3/4) = 1*3*(4/2) = 6.

Alternatively, if I think of it as 1*3*(4/2) = 1*3*2 = 6.

So, that seems to work.

Another test case: (2/3) @ (4/5). According to the definition, that would be 2*4*(5/3) = 8*(5/3) = 40/3 ≈ 13.333...

Alternatively, using the formula (m*p*q)/n: (2*4*5)/3 = 40/3. Correct.

So, that seems to check out.

Therefore, going back to the original problem, the answer is 49.

I think that's solid. I don't see any mistakes in my reasoning. So, I think the answer is 49.

**Final Answer**
\boxed{49}

To evaluate the expression $\frac{7}{30}@\frac{10}{21}$, we begin by understanding the operation $@$ as defined:

$$
\frac{m}{n} @ \frac{p}{q} = m \cdot p \cdot \left(\frac{q}{n}\right)
$$

This means we take the numerator of the first fraction ($m$), the numerator of the second fraction ($p$), and multiply them by the denominator of the second fraction ($q$) divided by the denominator of the first fraction ($n$).

---

### Step 1: Identify the components

From the given fractions:

- $\frac{7}{30}$ corresponds to $m = 7$, $n = 30$
- $\frac{10}{21}$ corresponds to $p = 10$, $q = 21$

Substitute into the operation:

$$
\frac{7}{30} @ \frac{10}{21} = 7 \cdot 10 \cdot \left(\frac{21}{30}\right)
$$

---

### Step 2: Simplify the expression

First, compute the product of the numerators:

$$
7 \cdot 10 = 70
$$

Now multiply by $\frac{21}{30}$:

$$
70 \cdot \frac{21}{30}
$$

We can simplify this by canceling common factors before multiplying:

- $70 = 7 \cdot 10$
- $30 = 3 \cdot 10$
- So, $\frac{70}{30} = \frac{7}{3}$

Now compute:

$$
\frac{7}{3} \cdot 21 = \frac{7 \cdot 21}{3} = \frac{147}{3} = 49
$$

---

### Final Answer

$$
\boxed{49}
$$

Solution: We have $\frac{7}{30}@\frac{10}{21}=(7)(10)\left(\frac{21}{30}\right)=\boxed{49}$.


Explanation:
Okay, so I need to figure out where the point (0,0) ends up after being reflected over the vertical line x=1 and then over the line y=2. Let me start by recalling what reflection over a vertical line does.

First, reflecting a point over a vertical line x = a. I think the formula for that is something like if you have a point (x, y), its reflection over x = a would be (2a - x, y). Let me check that. If the original point is to the left of the line, then the reflection should be the same distance on the other side. For example, if I reflect (0,0) over x=1, then the distance from 0 to 1 is 1 unit, so the reflection should be 1 unit to the right of x=1, which would be x=2. So the reflected point would be (2, 0). That seems right. So applying that formula, 2a - x where a is 1 and x is 0, so 2*1 - 0 = 2. Yep, that gives (2, 0).

Okay, so first reflection takes (0,0) to (2, 0). Now, the next step is reflecting that image over the line y=2. So now I need to figure out how to reflect a point over a horizontal line y = b. I think the formula is similar, but for the y-coordinate. If you have a point (x, y), its reflection over y = b would be (x, 2b - y). Let me verify. If the original point is below the line, then the reflection should be the same distance above. For example, reflecting (0,0) over y=2 would be 2 units up from y=2, which is y=4. So the reflected point would be (0, 4). Using the formula, 2*2 - 0 = 4, so (0, 4). That checks out.

So applying that to the point (2, 0) which we got after the first reflection. So reflecting over y=2, the new y-coordinate would be 2*2 - 0 = 4. Therefore, the resulting point is (2, 4).

Wait, let me make sure I didn't mix up anything. Let me go through each step again.

Original point: (0, 0).

First reflection over x=1. The formula for reflection over vertical line x=a is (2a - x, y). So substituting a=1, x=0, y=0: 2*1 - 0 = 2, so (2, 0). That seems correct.

Then reflecting (2, 0) over y=2. The formula for reflection over horizontal line y=b is (x, 2b - y). So substituting b=2, x=2, y=0: 2*2 - 0 = 4. So the new point is (2, 4).

Hmm, that seems straightforward. But maybe I should visualize it or check with another method.

Alternatively, thinking about reflections: reflecting over x=1. The point (0,0) is 1 unit to the left of x=1, so the reflection should be 1 unit to the right, which is (2, 0). Then reflecting (2, 0) over y=2. The point (2, 0) is 2 units below y=2, so the reflection should be 2 units above y=2, which is y=4. Therefore, (2, 4).

Yes, that matches. So maybe that's the answer.

But let me think if there's another way to approach this. Maybe using coordinate geometry formulas.

Alternatively, maybe using matrices or something? But reflections are usually straightforward with formulas.

Alternatively, think about the composition of reflections. But since the first reflection is over a vertical line and the second over a horizontal line, they are perpendicular. But maybe that's complicating.

Alternatively, think about the distance from the line. For reflection over x=1, the distance from (0,0) to x=1 is |0 - 1| = 1. So the reflection is on the other side, same distance, so 1 + 1 = 2. So x-coordinate is 2.

Similarly, for reflecting (2, 0) over y=2. The distance from (2, 0) to y=2 is |0 - 2| = 2. So the reflection is 2 units above y=2, so y-coordinate is 2 + 2 = 4.

Therefore, the final point is (2, 4).

Wait, so all methods lead to (2, 4). Is there any chance I made a mistake here? Let me check with coordinates.

Let me draw a rough sketch mentally. Original point is at the origin. Reflecting over x=1, which is a vertical line one unit to the right. So the reflection is at (2, 0). Then reflecting that over y=2, which is a horizontal line two units up. The point (2, 0) is two units below y=2, so reflecting over y=2 would take it to (2, 4).

Yes, that seems correct.

Alternatively, if I had messed up the order of reflections? Wait, the problem says first reflect over x=1, then over y=2. So the order is important. If I had done the other way around, reflecting over y=2 first, then over x=1, the result would be different. But here, the order is correct.

Alternatively, maybe I can check with another point. Let me take a different point to see if my reflection formulas are correct.

Suppose I reflect (3, 5) over x=1. According to the formula, it should be (2*1 - 3, 5) = (2 - 3, 5) = (-1, 5). Let me check. The distance from x=1 to x=3 is 2 units. So reflection should be 2 units to the left of x=1, which is x=1 - 2 = -1. So yes, (-1, 5). Correct.

Similarly, reflecting (-1, 5) over y=2. The formula would be ( -1, 2*2 - 5 ) = (-1, 4 - 5) = (-1, -1). Let me check. The distance from y=5 to y=2 is 3 units. So reflection should be 3 units below y=2, which is y=2 - 3 = -1. So yes, (-1, -1). Correct.

So the formulas are working. Therefore, applying them to the original problem, the answer is (2, 4).

Alternatively, maybe I can think of the composition of two reflections. But since reflections over vertical and horizontal lines are not the same as a single reflection, but rather a combination. However, since they are over different axes, the composition is a reflection over some line? Wait, no. Wait, reflecting over two lines that are perpendicular would result in a rotation. Wait, but here we have two reflections: first over vertical, then over horizontal.

But maybe that's more complicated. However, since we already have the answer through direct computation, maybe it's not necessary. But just for my understanding, let me think.

If you reflect over x=1 and then over y=2, what is the composition?

In general, reflecting over two lines that are not parallel results in a rotation. Since x=1 and y=2 are perpendicular, the composition is a rotation by 180 degrees around the point which is the intersection of the two lines? Wait, but the two lines x=1 and y=2 are perpendicular, but they don't intersect each other? Wait, no, they do intersect at (1, 2). Wait, x=1 is a vertical line, y=2 is a horizontal line, so they intersect at (1, 2).

So reflecting over x=1 and then over y=2 would be equivalent to a rotation of 180 degrees around the point (1, 2). Let me check that.

If that's the case, then applying a 180-degree rotation around (1, 2) to the original point (0, 0) would give us the final point.

How does a 180-degree rotation around a point work? The formula is: for a point (x, y), rotating 180 degrees around (h, k) gives (2h - x, 2k - y).

So applying that here, h=1, k=2. So the rotated point would be (2*1 - 0, 2*2 - 0) = (2, 4). Which is the same result as before!

So that's another way to see it. Therefore, the answer is (2, 4).

Therefore, regardless of the method, the answer is (2, 4).

But just to make sure, let me think again. If I have a 180-degree rotation around (1, 2), then the original point (0, 0) is being rotated. The vector from (1, 2) to (0, 0) is (-1, -2). Rotating that 180 degrees would give (1, 2) + (-1, -2) rotated 180, which is (1, 2) + (1, 2) = (2, 4). Yep, that works.

So that's another confirmation.

Alternatively, if I didn't know about the composition of reflections, but just did the two reflections step by step, I still get the same answer.

Therefore, I think I can be confident that the answer is (2, 4).

**Final Answer**
The resulting point is \boxed{(2, 4)}.

To determine the final position of the point $(0, 0)$ after two successive reflections, we proceed step by step, applying the rules of reflection over vertical and horizontal lines.

---

### **Step 1: Reflect over the vertical line $x = 1$**

The formula for reflecting a point $(x, y)$ over a vertical line $x = a$ is:

$$
(x', y') = (2a - x, y)
$$

Applying this to the point $(0, 0)$ with $a = 1$:

$$
x' = 2(1) - 0 = 2, \quad y' = 0
$$

So, the image after the first reflection is:

$$
(2, 0)
$$

---

### **Step 2: Reflect over the horizontal line $y = 2$**

The formula for reflecting a point $(x, y)$ over a horizontal line $y = b$ is:

$$
(x', y') = (x, 2b - y)
$$

Applying this to the point $(2, 0)$ with $b = 2$:

$$
x' = 2, \quad y' = 2(2) - 0 = 4
$$

So, the image after the second reflection is:

$$
(2, 4)
$$

---

### **Alternative Perspective: Composition of Reflections**

Reflecting over two perpendicular lines (a vertical and a horizontal) results in a **180° rotation** about the point of intersection of the two lines. The lines $x = 1$ and $y = 2$ intersect at the point $(1, 2)$.

A 180° rotation about a point $(h, k)$ transforms a point $(x, y)$ to:

$$
(2h - x, 2k - y)
$$

Applying this to the original point $(0, 0)$ with $(h, k) = (1, 2)$:

$$
x' = 2(1) - 0 = 2, \quad y' = 2(2) - 0 = 4
$$

This again gives the final point:

$$
(2, 4)
$$

---

### **Conclusion**

Both the step-by-step reflection method and the geometric interpretation of the composition of reflections lead to the same result.

$$
\boxed{(2, 4)}
$$

[Assistant]
Explanation:
Okay, so I need to figure out where the point (0,0) ends up after being reflected over the vertical line x=1 and then over the line y=2. Let me start by recalling what reflection over a vertical line does.

First, reflecting a point over a vertical line x = a. I think the formula for that is something like if you have a point (x, y), its reflection over x = a would be (2a - x, y). Let me check that. If the original point is to the left of the line, then the reflection should be the same distance on the other side. For example, if I reflect (0,0) over x=1, then the distance from 0 to 1 is 1 unit, so the reflection should be 1 unit to the right of x=1, which would be x=2. So the reflected point would be (2, 0). That seems right. So applying that formula, 2a - x where a is 1 and x is 0, so 2*1 - 0 = 2. Yep, that gives (2, 0).

Okay, so first reflection takes (0,0) to (2, 0). Now, the next step is reflecting that image over the line y=2. So now I need to figure out how to reflect a point over a horizontal line y = b. I think the formula is similar, but for the y-coordinate. If you have a point (x, y), its reflection over y = b would be (x, 2b - y). Let me verify. If the original point is below the line, then the reflection should be the same distance above. For example, reflecting (0,0) over y=2 would be 2 units up from y=2, which is y=4. So the reflected point would be (0, 4). Using the formula, 2*2 - 0 = 4, so (0, 4). That checks out.

So applying that to the point (2, 0) which we got after the first reflection. So reflecting over y=2, the new y-coordinate would be 2*2 - 0 = 4. Therefore, the resulting point is (2, 4).

Wait, let me make sure I didn't mix up anything. Let me go through each step again.

Original point: (0, 0).

First reflection over x=1. The formula for reflection over vertical line x=a is (2a - x, y). So substituting a=1, x=0, y=0: 2*1 - 0 = 2, so (2, 0). That seems correct.

Then reflecting (2, 0) over y=2. The formula for reflection over horizontal line y=b is (x, 2b - y). So substituting b=2, x=2, y=0: 2*2 - 0 = 4. So the new point is (2, 4).

Hmm, that seems straightforward. But maybe I should visualize it or check with another method.

Alternatively, thinking about reflections: reflecting over x=1. The point (0,0) is 1 unit to the left of x=1, so the reflection should be 1 unit to the right, which is (2, 0). Then reflecting (2, 0) over y=2. The point (2, 0) is 2 units below y=2, so the reflection should be 2 units above y=2, which is y=4. Therefore, (2, 4).

Yes, that matches. So maybe that's the answer.

But let me think if there's another way to approach this. Maybe using coordinate geometry formulas.

Alternatively, maybe using matrices or something? But reflections are usually straightforward with formulas.

Alternatively, think about the composition of reflections. But since the first reflection is over a vertical line and the second over a horizontal line, they are perpendicular. But maybe that's complicating.

Alternatively, think about the distance from the line. For reflection over x=1, the distance from (0,0) to x=1 is |0 - 1| = 1. So the reflection is on the other side, same distance, so 1 + 1 = 2. So x-coordinate is 2.

Similarly, for reflecting (2, 0) over y=2. The distance from (2, 0) to y=2 is |0 - 2| = 2. So the reflection is 2 units above y=2, so y-coordinate is 2 + 2 = 4.

Therefore, the final point is (2, 4).

Wait, so all methods lead to (2, 4). Is there any chance I made a mistake here? Let me check with coordinates.

Let me draw a rough sketch mentally. Original point is at the origin. Reflecting over x=1, which is a vertical line one unit to the right. So the reflection is at (2, 0). Then reflecting that over y=2, which is a horizontal line two units up. The point (2, 0) is two units below y=2, so reflecting over y=2 would take it to (2, 4).

Yes, that seems correct.

Alternatively, if I had messed up the order of reflections? Wait, the problem says first reflect over x=1, then over y=2. So the order is important. If I had done the other way around, reflecting over y=2 first, then over x=1, the result would be different. But here, the order is correct.

Alternatively, maybe I can check with another point. Let me take a different point to see if my reflection formulas are correct.

Suppose I reflect (3, 5) over x=1. According to the formula, it should be (2*1 - 3, 5) = (2 - 3, 5) = (-1, 5). Let me check. The distance from x=1 to x=3 is 2 units. So reflection should be 2 units to the left of x=1, which is x=1 - 2 = -1. So yes, (-1, 5). Correct.

Similarly, reflecting (-1, 5) over y=2. The formula would be ( -1, 2*2 - 5 ) = (-1, 4 - 5) = (-1, -1). Let me check. The distance from y=5 to y=2 is 3 units. So reflection should be 3 units below y=2, which is y=2 - 3 = -1. So yes, (-1, -1). Correct.

So the formulas are working. Therefore, applying them to the original problem, the answer is (2, 4).

Alternatively, maybe I can think of the composition of two reflections. But since reflections over vertical and horizontal lines are not the same as a single reflection, but rather a combination. However, since they are over different axes, the composition is a reflection over some line? Wait, no. Wait, reflecting over two lines that are perpendicular would result in a rotation. Wait, but here we have two reflections: first over vertical, then over horizontal.

But maybe that's more complicated. However, since we already have the answer through direct computation, maybe it's not necessary. But just for my understanding, let me think.

If you reflect over x=1 and then over y=2, what is the composition?

In general, reflecting over two lines that are not parallel results in a rotation. Since x=1 and y=2 are perpendicular, the composition is a rotation by 180 degrees around the point which is the intersection of the two lines? Wait, but the two lines x=1 and y=2 are perpendicular, but they don't intersect each other? Wait, no, they do intersect at (1, 2). Wait, x=1 is a vertical line, y=2 is a horizontal line, so they intersect at (1, 2).

So reflecting over x=1 and then over y=2 would be equivalent to a rotation of 180 degrees around the point (1, 2). Let me check that.

If that's the case, then applying a 180-degree rotation around (1, 2) to the original point (0, 0) would give us the final point.

How does a 180-degree rotation around a point work? The formula is: for a point (x, y), rotating 180 degrees around (h, k) gives (2h - x, 2k - y).

So applying that here, h=1, k=2. So the rotated point would be (2*1 - 0, 2*2 - 0) = (2, 4). Which is the same result as before!

So that's another way to see it. Therefore, the answer is (2, 4).

Therefore, regardless of the method, the answer is (2, 4).

But just to make sure, let me think again. If I have a 180-degree rotation around (1, 2), then the original point (0, 0) is being rotated. The vector from (1, 2) to (0, 0) is (-1, -2). Rotating that 180 degrees would give (1, 2) + (-1, -2) rotated 180, which is (1, 2) + (1, 2) = (2, 4). Yep, that works.

So that's another confirmation.

Alternatively, if I didn't know about the composition of reflections, but just did the two reflections step by step, I still get the same answer.

Therefore, I think I can be confident that the answer is (2, 4).

**Final Answer**
The resulting point is \boxed{(2, 4)}.

To determine the final position of the point $(0, 0)$ after two successive reflections, we proceed step by step, applying the rules of reflection over vertical and horizontal lines.

---

### **Step 1: Reflect over the vertical line $x = 1$**

The formula for reflecting a point $(x, y)$ over a vertical line $x = a$ is:

$$
(x', y') = (2a - x, y)
$$

Applying this to the point $(0, 0)$ with $a = 1$:

$$
x' = 2(1) - 0 = 2, \quad y' = 0
$$

So, the image after the first reflection is:

$$
(2, 0)
$$

---

### **Step 2: Reflect over the horizontal line $y = 2$**

The formula for reflecting a point $(x, y)$ over a horizontal line $y = b$ is:

$$
(x', y') = (x, 2b - y)
$$

Applying this to the point $(2, 0)$ with $b = 2$:

$$
x' = 2, \quad y' = 2(2) - 0 = 4
$$

So, the image after the second reflection is:

$$
(2, 4)
$$

---

### **Alternative Perspective: Composition of Reflections**

Reflecting over two perpendicular lines (a vertical and a horizontal) results in a **180° rotation** about the point of intersection of the two lines. The lines $x = 1$ and $y = 2$ intersect at the point $(1, 2)$.

A 180° rotation about a point $(h, k)$ transforms a point $(x, y)$ to:

$$
(2h - x, 2k - y)
$$

Applying this to the original point $(0, 0)$ with $(h, k) = (1, 2)$:

$$
x' = 2(1) - 0 = 2, \quad y' = 2(2) - 0 = 4
$$

This again gives the final point:

$$
(2, 4)
$$

---

### **Conclusion**

Both the step-by-step reflection method and the geometric interpretation of the composition of reflections lead to the same result.

$$
\boxed{(2, 4)}
$$

Solution:
When $(0,0)$ is reflected over the line $x=1$, the image is $(2,0)$. [asy]
draw((-2, 0)--(6, 0), Arrow); draw((0, -2)--(0, 6), Arrow);
label("$x$", (6, 0), E); label("$y$", (0, 6), N);
label("$(0, 0)$", (0, 0), SW); label("$(2, 0)$", (2, 0), SE);
label("$(2, 4)$", (2, 4), NE);
label("$y = 2$", (6, 2), E); label("$x = 1$", (1, -2), S);
draw((-2, 2)--(6, 2), dashed); draw((1, -2)--(1, 6), dashed);
dot((0, 0)); dot((2, 0)); dot((2, 4));
[/asy] When $(2,0)$ is reflected over the line $y=2$, the image is $\boxed{(2,4)}$.


Explanation:
Okay, so I need to factor the quadratic expression $10x^2 - x - 24$ into the form $(Ax - 8)(Bx + 3)$, where A and B are integers. Then, I have to find the value of $AB + B$. Alright, let me think about how to approach this.

First, I remember that factoring a quadratic in the form $ax^2 + bx + c$ usually involves finding two binomials $(mx + n)(px + q)$ such that $m \times p = a$, $n \times q = c$, and the middle term $bx$ comes from the sum of the products of the outer and inner terms. But in this case, the problem already gives me the form $(Ax - 8)(Bx + 3)$, so maybe I can use that structure to find A and B.

Let me write down the given expression and the factored form:

Original expression: $10x^2 - x - 24$

Factored form: $(Ax - 8)(Bx + 3)$

First, let me expand the factored form to see what it equals:

$(Ax - 8)(Bx + 3) = Ax \times Bx + Ax \times 3 - 8 \times Bx - 8 \times 3$

Calculating each term:

- First term: $A \times B \times x^2 = ABx^2$
- Second term: $3A x$
- Third term: $-8B x$
- Fourth term: $-24$

So combining like terms:

$ABx^2 + (3A - 8B)x - 24$

Now, this should be equal to the original expression $10x^2 - x - 24$. Therefore, we can set up equations by matching the coefficients of corresponding terms.

So, for the $x^2$ term:

$AB = 10$  ...(1)

For the $x$ term:

$3A - 8B = -1$  ...(2)

For the constant term:

$-24 = -24$  ...(3)

Equation (3) is already satisfied, so we don't need to worry about that. Now, we have two equations (1) and (2) with two variables A and B. Let me write them again:

1. $AB = 10$
2. $3A - 8B = -1$

Our goal is to find integers A and B that satisfy these equations. Let me think about how to solve this system.

Since AB = 10, and A and B are integers, we can list the possible integer pairs (A, B) that multiply to 10. Let me recall that 10 can be factored into integers as:

(1, 10), (2, 5), (5, 2), (10, 1), (-1, -10), (-2, -5), (-5, -2), (-10, -1)

So these are all the possible pairs. Now, we need to check which of these pairs satisfy the second equation $3A - 8B = -1$.

Let me go through each pair one by one.

First pair: (1, 10)

Check equation 2: 3(1) - 8(10) = 3 - 80 = -77 ≠ -1. Not good.

Second pair: (2, 5)

3(2) - 8(5) = 6 - 40 = -34 ≠ -1. Nope.

Third pair: (5, 2)

3(5) - 8(2) = 15 - 16 = -1. Hey, that works! So this pair (5, 2) satisfies equation 2. Let me check if there are other pairs that might also work.

Fourth pair: (10, 1)

3(10) - 8(1) = 30 - 8 = 22 ≠ -1. Not good.

Fifth pair: (-1, -10)

3(-1) - 8(-10) = -3 + 80 = 77 ≠ -1. Nope.

Sixth pair: (-2, -5)

3(-2) - 8(-5) = -6 + 40 = 34 ≠ -1. Not.

Seventh pair: (-5, -2)

3(-5) - 8(-2) = -15 + 16 = 1 ≠ -1. Close, but not quite.

Eighth pair: (-10, -1)

3(-10) - 8(-1) = -30 + 8 = -22 ≠ -1. Nope.

So only the third pair (5, 2) works. Therefore, A = 5 and B = 2.

Wait, but let me double-check. Let me plug A = 5 and B = 2 into the original factored form and see if it gives the original expression.

So, (5x - 8)(2x + 3). Let me expand this:

First, multiply 5x by 2x: 10x²

Then, 5x * 3 = 15x

Then, -8 * 2x = -16x

Then, -8 * 3 = -24

Combine the middle terms: 15x -16x = -x

So altogether, 10x² - x -24. Perfect! That's exactly the original expression. So A = 5 and B = 2.

Therefore, AB + B is (5*2) + 2 = 10 + 2 = 12. Wait, but let me check if there are other possible pairs. Wait, earlier when I checked the pairs, only (5, 2) worked. But let me think again. Maybe I missed something?

Wait, AB = 10. So A and B can be positive or negative. But in the pairs I considered, I included all possible integer pairs. So (5, 2) is the only one that works. Therefore, AB + B is 5*2 + 2 = 12. So the answer is 12?

Wait, but let me check again. Let me think if there's another way to approach this problem, maybe using the method of factoring quadratics.

Alternatively, maybe I can use the method of splitting the middle term. Let me try that as a check.

Given the quadratic $10x^2 - x -24$. To factor this, we need two numbers that multiply to $10 * (-24) = -240$ and add up to -1. Let me find such numbers.

Looking for two numbers m and n such that m * n = -240 and m + n = -1.

So, since their product is negative, one is positive and the other is negative. Also, their sum is -1, which is negative. Therefore, the larger number in absolute value is negative, and the smaller is positive. Let me think.

Let me think of factors of 240. Let me list pairs of factors of 240:

1 and 240

2 and 120

3 and 80

4 and 60

5 and 48

6 and 40

8 and 30

10 and 24

12 and 20

15 and 16

Now, since we need one positive and one negative, and their sum is -1. Let me think. Let me suppose that the numbers are m and n, with m positive and n negative. Then m - |n| = -1. Also, m * |n| = 240.

So m - |n| = -1 => m = |n| -1. Then substituting into m * |n| = 240:

(|n| -1) * |n| = 240 => |n|² - |n| - 240 = 0.

Solving quadratic equation for |n|:

|n| = [1 ± sqrt(1 + 960)] / 2 = [1 ± sqrt(961)] / 2 = [1 ± 31]/2

So positive solution: (1 + 31)/2 = 32/2 = 16. The other solution is negative, which we can ignore since |n| is positive.

Therefore, |n| = 16, so m = 16 -1 = 15. Therefore, the numbers are m = 15 and n = -16. Let me check: 15 * (-16) = -240, and 15 + (-16) = -1. Perfect!

Therefore, we can split the middle term -x as 15x -16x. Therefore, the quadratic becomes:

$10x^2 +15x -16x -24$

Now, group the terms:

(10x² +15x) + (-16x -24)

Factor out common terms:

From first group: 5x(2x + 3)

From second group: -8(2x + 3)

So, we have 5x(2x + 3) -8(2x + 3) = (5x -8)(2x + 3)

Which is the same as (Ax -8)(Bx +3) with A=5 and B=2. So same result as before. Therefore, AB + B = 5*2 + 2 = 12. So that's consistent.

Therefore, the answer is 12. Wait, but let me check once more if there are other possibilities. For example, if I had considered different factor pairs for AB=10. Wait, but we already checked all possible integer pairs. So (5,2) is the only one that works. Therefore, the answer is 12.

But just to be thorough, let me check if there's another way. Suppose I didn't think of splitting the middle term. Let me try solving equations (1) and (2) algebraically.

From equation (1): AB = 10. So A = 10/B. Then substitute into equation (2):

3*(10/B) -8B = -1

Multiply both sides by B to eliminate denominator:

3*10 -8B² = -B

Which is:

30 -8B² = -B

Bring all terms to one side:

-8B² + B + 30 = 0

Multiply both sides by -1:

8B² - B -30 = 0

Now, solving quadratic equation 8B² - B -30 = 0.

Using quadratic formula: B = [1 ± sqrt(1 + 4*8*30)] / (2*8)

Calculate discriminant:

sqrt(1 + 960) = sqrt(961) = 31

Therefore, B = [1 ± 31]/16

So two solutions:

B = (1 + 31)/16 = 32/16 = 2

B = (1 - 31)/16 = (-30)/16 = -15/8

But since B must be integer, only B=2 is valid. Therefore, B=2. Then from equation (1), AB=10 => A=10/2=5. Therefore, A=5, B=2. Same result. So AB + B = 5*2 + 2 = 12. So that's consistent.

Therefore, regardless of the method, the answer is 12. Therefore, I can be confident that the answer is 12.

But just to make sure, let me check if there's any other possible factor pairs for AB=10 that I might have missed. For example, if A and B are fractions? But the problem states that A and B are integers, so fractions are not allowed. So only integer pairs. So the pairs I considered earlier are all. So yes, only (5,2) works.

Alternatively, maybe there's a different way to factor? Let me think. Suppose someone tried different signs. Wait, but in the original problem, the factored form is (Ax -8)(Bx +3). So the constants are -8 and +3. So when we expanded, we had -8*Bx and +3*A x. So the middle term is 3A -8B. Which we set equal to -1. So that's correct.

Alternatively, if someone thought maybe the factors are (Ax + something)(Bx + something else), but the problem specifies the form (Ax -8)(Bx +3). So the constants are fixed as -8 and +3. Therefore, the rest of the coefficients have to adjust accordingly. So our approach is correct.

Therefore, I think the answer is 12. So AB + B = 5*2 + 2 = 12. Therefore, the answer is 12.

**Final Answer**
\boxed{12}

To solve the problem, we are given the quadratic expression:

$$
10x^2 - x - 24
$$

and asked to express it in the form:

$$
(Ax - 8)(Bx + 3)
$$

where $ A $ and $ B $ are integers. Then, we are to compute $ AB + B $.

---

### Step 1: Expand the Factored Form

We expand the given factored form:

$$
(Ax - 8)(Bx + 3) = ABx^2 + 3Ax - 8Bx - 24
$$

Grouping like terms:

$$
ABx^2 + (3A - 8B)x - 24
$$

Now, compare this with the original expression:

$$
10x^2 - x - 24
$$

Matching coefficients:

- Coefficient of $ x^2 $: $ AB = 10 $
- Coefficient of $ x $: $ 3A - 8B = -1 $
- Constant term: $ -24 = -24 $ (already matches)

---

### Step 2: Solve the System of Equations

We now solve the system:

1. $ AB = 10 $
2. $ 3A - 8B = -1 $

From equation (1), we can list all integer pairs $ (A, B) $ such that $ AB = 10 $:

$$
(1, 10), (2, 5), (5, 2), (10, 1), (-1, -10), (-2, -5), (-5, -2), (-10, -1)
$$

Now, substitute each pair into equation (2) to find which one satisfies $ 3A - 8B = -1 $:

- $ (1, 10): 3(1) - 8(10) = 3 - 80 = -77 $ ❌
- $ (2, 5): 3(2) - 8(5) = 6 - 40 = -34 $ ❌
- $ (5, 2): 3(5) - 8(2) = 15 - 16 = -1 $ ✅
- $ (10, 1): 3(10) - 8(1) = 30 - 8 = 22 $ ❌
- $ (-1, -10): 3(-1) - 8(-10) = -3 + 80 = 77 $ ❌
- $ (-2, -5): 3(-2) - 8(-5) = -6 + 40 = 34 $ ❌
- $ (-5, -2): 3(-5) - 8(-2) = -15 + 16 = 1 $ ❌
- $ (-10, -1): 3(-10) - 8(-1) = -30 + 8 = -22 $ ❌

Only the pair $ (A, B) = (5, 2) $ satisfies both equations.

---

### Step 3: Compute $ AB + B $

Now that we have $ A = 5 $ and $ B = 2 $, compute:

$$
AB + B = (5)(2) + 2 = 10 + 2 = \boxed{12}
$$

---

### Final Answer

$$
\boxed{12}
$$

[Assistant]
Explanation:
Okay, so I need to factor the quadratic expression $10x^2 - x - 24$ into the form $(Ax - 8)(Bx + 3)$, where A and B are integers. Then, I have to find the value of $AB + B$. Alright, let me think about how to approach this.

First, I remember that factoring a quadratic in the form $ax^2 + bx + c$ usually involves finding two binomials $(mx + n)(px + q)$ such that $m \times p = a$, $n \times q = c$, and the middle term $bx$ comes from the sum of the products of the outer and inner terms. But in this case, the problem already gives me the form $(Ax - 8)(Bx + 3)$, so maybe I can use that structure to find A and B.

Let me write down the given expression and the factored form:

Original expression: $10x^2 - x - 24$

Factored form: $(Ax - 8)(Bx + 3)$

First, let me expand the factored form to see what it equals:

$(Ax - 8)(Bx + 3) = Ax \times Bx + Ax \times 3 - 8 \times Bx - 8 \times 3$

Calculating each term:

- First term: $A \times B \times x^2 = ABx^2$
- Second term: $3A x$
- Third term: $-8B x$
- Fourth term: $-24$

So combining like terms:

$ABx^2 + (3A - 8B)x - 24$

Now, this should be equal to the original expression $10x^2 - x - 24$. Therefore, we can set up equations by matching the coefficients of corresponding terms.

So, for the $x^2$ term:

$AB = 10$  ...(1)

For the $x$ term:

$3A - 8B = -1$  ...(2)

For the constant term:

$-24 = -24$  ...(3)

Equation (3) is already satisfied, so we don't need to worry about that. Now, we have two equations (1) and (2) with two variables A and B. Let me write them again:

1. $AB = 10$
2. $3A - 8B = -1$

Our goal is to find integers A and B that satisfy these equations. Let me think about how to solve this system.

Since AB = 10, and A and B are integers, we can list the possible integer pairs (A, B) that multiply to 10. Let me recall that 10 can be factored into integers as:

(1, 10), (2, 5), (5, 2), (10, 1), (-1, -10), (-2, -5), (-5, -2), (-10, -1)

So these are all the possible pairs. Now, we need to check which of these pairs satisfy the second equation $3A - 8B = -1$.

Let me go through each pair one by one.

First pair: (1, 10)

Check equation 2: 3(1) - 8(10) = 3 - 80 = -77 ≠ -1. Not good.

Second pair: (2, 5)

3(2) - 8(5) = 6 - 40 = -34 ≠ -1. Nope.

Third pair: (5, 2)

3(5) - 8(2) = 15 - 16 = -1. Hey, that works! So this pair (5, 2) satisfies equation 2. Let me check if there are other pairs that might also work.

Fourth pair: (10, 1)

3(10) - 8(1) = 30 - 8 = 22 ≠ -1. Not good.

Fifth pair: (-1, -10)

3(-1) - 8(-10) = -3 + 80 = 77 ≠ -1. Nope.

Sixth pair: (-2, -5)

3(-2) - 8(-5) = -6 + 40 = 34 ≠ -1. Not.

Seventh pair: (-5, -2)

3(-5) - 8(-2) = -15 + 16 = 1 ≠ -1. Close, but not quite.

Eighth pair: (-10, -1)

3(-10) - 8(-1) = -30 + 8 = -22 ≠ -1. Nope.

So only the third pair (5, 2) works. Therefore, A = 5 and B = 2.

Wait, but let me double-check. Let me plug A = 5 and B = 2 into the original factored form and see if it gives the original expression.

So, (5x - 8)(2x + 3). Let me expand this:

First, multiply 5x by 2x: 10x²

Then, 5x * 3 = 15x

Then, -8 * 2x = -16x

Then, -8 * 3 = -24

Combine the middle terms: 15x -16x = -x

So altogether, 10x² - x -24. Perfect! That's exactly the original expression. So A = 5 and B = 2.

Therefore, AB + B is (5*2) + 2 = 10 + 2 = 12. Wait, but let me check if there are other possible pairs. Wait, earlier when I checked the pairs, only (5, 2) worked. But let me think again. Maybe I missed something?

Wait, AB = 10. So A and B can be positive or negative. But in the pairs I considered, I included all possible integer pairs. So (5, 2) is the only one that works. Therefore, AB + B is 5*2 + 2 = 12. So the answer is 12?

Wait, but let me check again. Let me think if there's another way to approach this problem, maybe using the method of factoring quadratics.

Alternatively, maybe I can use the method of splitting the middle term. Let me try that as a check.

Given the quadratic $10x^2 - x -24$. To factor this, we need two numbers that multiply to $10 * (-24) = -240$ and add up to -1. Let me find such numbers.

Looking for two numbers m and n such that m * n = -240 and m + n = -1.

So, since their product is negative, one is positive and the other is negative. Also, their sum is -1, which is negative. Therefore, the larger number in absolute value is negative, and the smaller is positive. Let me think.

Let me think of factors of 240. Let me list pairs of factors of 240:

1 and 240

2 and 120

3 and 80

4 and 60

5 and 48

6 and 40

8 and 30

10 and 24

12 and 20

15 and 16

Now, since we need one positive and one negative, and their sum is -1. Let me think. Let me suppose that the numbers are m and n, with m positive and n negative. Then m - |n| = -1. Also, m * |n| = 240.

So m - |n| = -1 => m = |n| -1. Then substituting into m * |n| = 240:

(|n| -1) * |n| = 240 => |n|² - |n| - 240 = 0.

Solving quadratic equation for |n|:

|n| = [1 ± sqrt(1 + 960)] / 2 = [1 ± sqrt(961)] / 2 = [1 ± 31]/2

So positive solution: (1 + 31)/2 = 32/2 = 16. The other solution is negative, which we can ignore since |n| is positive.

Therefore, |n| = 16, so m = 16 -1 = 15. Therefore, the numbers are m = 15 and n = -16. Let me check: 15 * (-16) = -240, and 15 + (-16) = -1. Perfect!

Therefore, we can split the middle term -x as 15x -16x. Therefore, the quadratic becomes:

$10x^2 +15x -16x -24$

Now, group the terms:

(10x² +15x) + (-16x -24)

Factor out common terms:

From first group: 5x(2x + 3)

From second group: -8(2x + 3)

So, we have 5x(2x + 3) -8(2x + 3) = (5x -8)(2x + 3)

Which is the same as (Ax -8)(Bx +3) with A=5 and B=2. So same result as before. Therefore, AB + B = 5*2 + 2 = 12. So that's consistent.

Therefore, the answer is 12. Wait, but let me check once more if there are other possibilities. For example, if I had considered different factor pairs for AB=10. Wait, but we already checked all possible integer pairs. So (5,2) is the only one that works. Therefore, the answer is 12.

But just to be thorough, let me check if there's another way. Suppose I didn't think of splitting the middle term. Let me try solving equations (1) and (2) algebraically.

From equation (1): AB = 10. So A = 10/B. Then substitute into equation (2):

3*(10/B) -8B = -1

Multiply both sides by B to eliminate denominator:

3*10 -8B² = -B

Which is:

30 -8B² = -B

Bring all terms to one side:

-8B² + B + 30 = 0

Multiply both sides by -1:

8B² - B -30 = 0

Now, solving quadratic equation 8B² - B -30 = 0.

Using quadratic formula: B = [1 ± sqrt(1 + 4*8*30)] / (2*8)

Calculate discriminant:

sqrt(1 + 960) = sqrt(961) = 31

Therefore, B = [1 ± 31]/16

So two solutions:

B = (1 + 31)/16 = 32/16 = 2

B = (1 - 31)/16 = (-30)/16 = -15/8

But since B must be integer, only B=2 is valid. Therefore, B=2. Then from equation (1), AB=10 => A=10/2=5. Therefore, A=5, B=2. Same result. So AB + B = 5*2 + 2 = 12. So that's consistent.

Therefore, regardless of the method, the answer is 12. Therefore, I can be confident that the answer is 12.

But just to make sure, let me check if there's any other possible factor pairs for AB=10 that I might have missed. For example, if A and B are fractions? But the problem states that A and B are integers, so fractions are not allowed. So only integer pairs. So the pairs I considered earlier are all. So yes, only (5,2) works.

Alternatively, maybe there's a different way to factor? Let me think. Suppose someone tried different signs. Wait, but in the original problem, the factored form is (Ax -8)(Bx +3). So the constants are -8 and +3. So when we expanded, we had -8*Bx and +3*A x. So the middle term is 3A -8B. Which we set equal to -1. So that's correct.

Alternatively, if someone thought maybe the factors are (Ax + something)(Bx + something else), but the problem specifies the form (Ax -8)(Bx +3). So the constants are fixed as -8 and +3. Therefore, the rest of the coefficients have to adjust accordingly. So our approach is correct.

Therefore, I think the answer is 12. So AB + B = 5*2 + 2 = 12. Therefore, the answer is 12.

**Final Answer**
\boxed{12}

To solve the problem, we are given the quadratic expression:

$$
10x^2 - x - 24
$$

and asked to express it in the form:

$$
(Ax - 8)(Bx + 3)
$$

where $ A $ and $ B $ are integers. Then, we are to compute $ AB + B $.

---

### Step 1: Expand the Factored Form

We expand the given factored form:

$$
(Ax - 8)(Bx + 3) = ABx^2 + 3Ax - 8Bx - 24
$$

Grouping like terms:

$$
ABx^2 + (3A - 8B)x - 24
$$

Now, compare this with the original expression:

$$
10x^2 - x - 24
$$

Matching coefficients:

- Coefficient of $ x^2 $: $ AB = 10 $
- Coefficient of $ x $: $ 3A - 8B = -1 $
- Constant term: $ -24 = -24 $ (already matches)

---

### Step 2: Solve the System of Equations

We now solve the system:

1. $ AB = 10 $
2. $ 3A - 8B = -1 $

From equation (1), we can list all integer pairs $ (A, B) $ such that $ AB = 10 $:

$$
(1, 10), (2, 5), (5, 2), (10, 1), (-1, -10), (-2, -5), (-5, -2), (-10, -1)
$$

Now, substitute each pair into equation (2) to find which one satisfies $ 3A - 8B = -1 $:

- $ (1, 10): 3(1) - 8(10) = 3 - 80 = -77 $ ❌
- $ (2, 5): 3(2) - 8(5) = 6 - 40 = -34 $ ❌
- $ (5, 2): 3(5) - 8(2) = 15 - 16 = -1 $ ✅
- $ (10, 1): 3(10) - 8(1) = 30 - 8 = 22 $ ❌
- $ (-1, -10): 3(-1) - 8(-10) = -3 + 80 = 77 $ ❌
- $ (-2, -5): 3(-2) - 8(-5) = -6 + 40 = 34 $ ❌
- $ (-5, -2): 3(-5) - 8(-2) = -15 + 16 = 1 $ ❌
- $ (-10, -1): 3(-10) - 8(-1) = -30 + 8 = -22 $ ❌

Only the pair $ (A, B) = (5, 2) $ satisfies both equations.

---

### Step 3: Compute $ AB + B $

Now that we have $ A = 5 $ and $ B = 2 $, compute:

$$
AB + B = (5)(2) + 2 = 10 + 2 = \boxed{12}
$$

---

### Final Answer

$$
\boxed{12}
$$

Solution: We see that $10x^2-x-24=(5x-8)(2x+3)$, thus $A = 5$ and $B = 2$. Hence, $AB + B = \boxed{12}.$


Explanation:
Okay, so I need to combine these two fractions into a single fraction. The expression is (4 + 6a)/5 minus (1 + 3a)/4. Hmm, let me think. To combine fractions, they need a common denominator. The denominators here are 5 and 4. What's the least common denominator for 5 and 4? Well, 5 and 4 are both prime numbers, right? So their least common multiple would be 5 times 4, which is 20. So I need to convert both fractions to have 20 as the denominator.

Starting with the first fraction: (4 + 6a)/5. To get from 5 to 20, I multiply the denominator by 4. So I have to do the same to the numerator. That means multiplying (4 + 6a) by 4. Let me write that out: 4*(4 + 6a). Let me calculate that. 4 times 4 is 16, and 4 times 6a is 24a. So the first fraction becomes (16 + 24a)/20.

Now the second fraction is (1 + 3a)/4. To get the denominator from 4 to 20, I need to multiply by 5. So again, I have to multiply the numerator by 5. That would be 5*(1 + 3a). Let me compute that. 5 times 1 is 5, and 5 times 3a is 15a. So the second fraction becomes (5 + 15a)/20.

Now the original expression is the first fraction minus the second fraction, so it's (16 + 24a)/20 minus (5 + 15a)/20. Since the denominators are the same now, I can combine the numerators. So that would be [ (16 + 24a) - (5 + 15a) ] / 20.

Let me simplify the numerator. Distribute the negative sign to both terms in the second numerator. So that becomes 16 + 24a - 5 - 15a. Now combine like terms. Let's do the constants first: 16 - 5 is 11. Then the terms with a: 24a - 15a is 9a. So the numerator simplifies to 11 + 9a. Therefore, the combined fraction is (11 + 9a)/20.

Wait, let me check my steps again to make sure I didn't make a mistake. Starting with the first fraction: 4*(4 + 6a) is indeed 16 + 24a. Second fraction: 5*(1 + 3a) is 5 + 15a. Then subtracting the second fraction: 16 + 24a - 5 - 15a. Yes, 16 - 5 is 11, and 24a - 15a is 9a. So numerator is 11 + 9a over 20. That seems right.

Alternatively, maybe I can write it as (9a + 11)/20. Either way is fine, but maybe they prefer the terms in a certain order? Probably not important. Let me check if this can be simplified further. The numerator is 9a + 11. The denominator is 20. Are there any common factors between 9, 11, and 20? Well, 9 and 20 have a common factor of 1, 11 and 20 also have a common factor of 1. So the fraction is already in its simplest form.

Wait, but let me check if I did the signs correctly. The original problem was (4 + 6a)/5 - (1 + 3a)/4. When I converted them to denominators of 20, I had (16 + 24a)/20 - (5 + 15a)/20. So subtracting the entire second fraction. So yes, that's correct. So the subtraction applies to both terms in the numerator. So 16 - 5 and 24a - 15a. Correct.

Alternatively, maybe I can check by plugging in a value for a. Let me choose a simple value, like a = 0. Let's compute the original expression and the simplified one to see if they match.

Original expression when a = 0: (4 + 0)/5 - (1 + 0)/4 = 4/5 - 1/4. Let me compute that. 4/5 is 0.8, 1/4 is 0.25. 0.8 - 0.25 = 0.55. Now the simplified expression when a = 0 is (11 + 0)/20 = 11/20, which is 0.55. So that matches. Good.

Another check: let me pick a = 1. Original expression: (4 + 6*1)/5 - (1 + 3*1)/4 = (4 + 6)/5 - (1 + 3)/4 = 10/5 - 4/4 = 2 - 1 = 1. Simplified expression: (9*1 + 11)/20 = (9 + 11)/20 = 20/20 = 1. Perfect, that matches too.

Another check: a = 2. Original expression: (4 + 6*2)/5 - (1 + 3*2)/4 = (4 + 12)/5 - (1 + 6)/4 = 16/5 - 7/4. Let me compute that. 16/5 is 3.2, 7/4 is 1.75. 3.2 - 1.75 = 1.45. Simplified expression: (9*2 + 11)/20 = (18 + 11)/20 = 29/20 = 1.45. Correct again.

So seems like my answer is correct. Therefore, the expression as a single fraction is (9a + 11)/20. Alternatively, written as (11 + 9a)/20. Both are equivalent. Depending on the preferred order, but usually variables come first, so maybe 9a + 11 over 20.

Wait, but in the original problem, the first fraction is (4 + 6a)/5. So maybe they want the terms in the same order? But in the numerator, when we combined, it was 11 + 9a. But 9a + 11 is the same. So either way is acceptable.

Alternatively, maybe I can write it as (9a + 11)/20. Yeah, that's probably the standard way. So I think that's the answer.

Just to recap the steps:

1. Find common denominator: 20.
2. Convert each fraction:
   - (4 + 6a)/5 = (16 + 24a)/20
   - (1 + 3a)/4 = (5 + 15a)/20
3. Subtract the second fraction from the first:
   - (16 + 24a - 5 - 15a)/20
4. Combine like terms:
   - 16 - 5 = 11
   - 24a - 15a = 9a
5. Result: (11 + 9a)/20 or (9a + 11)/20.

All steps check out. So I think that's the correct answer.

**Final Answer**
The expression as a single fraction is \boxed{\dfrac{9a + 11}{20}}.

To simplify the expression
$$
\frac{4+6a}{5} - \frac{1+3a}{4}
$$
into a single fraction, we follow a systematic process of finding a common denominator and combining the numerators.

---

### **Step 1: Find the Least Common Denominator (LCD)**

The denominators are 5 and 4. Since they are relatively prime, their least common denominator is:

$$
\text{LCD} = 5 \times 4 = 20
$$

---

### **Step 2: Convert Each Fraction to Have the Common Denominator**

We rewrite each fraction with denominator 20:

- For $\frac{4 + 6a}{5}$, multiply numerator and denominator by 4:
  $$
  \frac{4 + 6a}{5} = \frac{4(4 + 6a)}{20} = \frac{16 + 24a}{20}
  $$

- For $\frac{1 + 3a}{4}$, multiply numerator and denominator by 5:
  $$
  \frac{1 + 3a}{4} = \frac{5(1 + 3a)}{20} = \frac{5 + 15a}{20}
  $$

---

### **Step 3: Subtract the Two Fractions**

Now that both fractions have the same denominator, we subtract the numerators:

$$
\frac{16 + 24a}{20} - \frac{5 + 15a}{20} = \frac{(16 + 24a) - (5 + 15a)}{20}
$$

Distribute the subtraction across the second numerator:

$$
= \frac{16 + 24a - 5 - 15a}{20}
$$

---

### **Step 4: Combine Like Terms**

Group and simplify the terms in the numerator:

- Constant terms: $16 - 5 = 11$
- Terms with $a$: $24a - 15a = 9a$

So the numerator becomes:

$$
11 + 9a
$$

---

### **Step 5: Write the Final Answer**

Now, the expression is a single fraction:

$$
\frac{11 + 9a}{20}
$$

This is the simplified form of the original expression.

---

### **Final Answer**

$$
\boxed{\dfrac{9a + 11}{20}}
$$

[Assistant]
Explanation:
Okay, so I need to combine these two fractions into a single fraction. The expression is (4 + 6a)/5 minus (1 + 3a)/4. Hmm, let me think. To combine fractions, they need a common denominator. The denominators here are 5 and 4. What's the least common denominator for 5 and 4? Well, 5 and 4 are both prime numbers, right? So their least common multiple would be 5 times 4, which is 20. So I need to convert both fractions to have 20 as the denominator.

Starting with the first fraction: (4 + 6a)/5. To get from 5 to 20, I multiply the denominator by 4. So I have to do the same to the numerator. That means multiplying (4 + 6a) by 4. Let me write that out: 4*(4 + 6a). Let me calculate that. 4 times 4 is 16, and 4 times 6a is 24a. So the first fraction becomes (16 + 24a)/20.

Now the second fraction is (1 + 3a)/4. To get the denominator from 4 to 20, I need to multiply by 5. So again, I have to multiply the numerator by 5. That would be 5*(1 + 3a). Let me compute that. 5 times 1 is 5, and 5 times 3a is 15a. So the second fraction becomes (5 + 15a)/20.

Now the original expression is the first fraction minus the second fraction, so it's (16 + 24a)/20 minus (5 + 15a)/20. Since the denominators are the same now, I can combine the numerators. So that would be [ (16 + 24a) - (5 + 15a) ] / 20.

Let me simplify the numerator. Distribute the negative sign to both terms in the second numerator. So that becomes 16 + 24a - 5 - 15a. Now combine like terms. Let's do the constants first: 16 - 5 is 11. Then the terms with a: 24a - 15a is 9a. So the numerator simplifies to 11 + 9a. Therefore, the combined fraction is (11 + 9a)/20.

Wait, let me check my steps again to make sure I didn't make a mistake. Starting with the first fraction: 4*(4 + 6a) is indeed 16 + 24a. Second fraction: 5*(1 + 3a) is 5 + 15a. Then subtracting the second fraction: 16 + 24a - 5 - 15a. Yes, 16 - 5 is 11, and 24a - 15a is 9a. So numerator is 11 + 9a over 20. That seems right.

Alternatively, maybe I can write it as (9a + 11)/20. Either way is fine, but maybe they prefer the terms in a certain order? Probably not important. Let me check if this can be simplified further. The numerator is 9a + 11. The denominator is 20. Are there any common factors between 9, 11, and 20? Well, 9 and 20 have a common factor of 1, 11 and 20 also have a common factor of 1. So the fraction is already in its simplest form.

Wait, but let me check if I did the signs correctly. The original problem was (4 + 6a)/5 - (1 + 3a)/4. When I converted them to denominators of 20, I had (16 + 24a)/20 - (5 + 15a)/20. So subtracting the entire second fraction. So yes, that's correct. So the subtraction applies to both terms in the numerator. So 16 - 5 and 24a - 15a. Correct.

Alternatively, maybe I can check by plugging in a value for a. Let me choose a simple value, like a = 0. Let's compute the original expression and the simplified one to see if they match.

Original expression when a = 0: (4 + 0)/5 - (1 + 0)/4 = 4/5 - 1/4. Let me compute that. 4/5 is 0.8, 1/4 is 0.25. 0.8 - 0.25 = 0.55. Now the simplified expression when a = 0 is (11 + 0)/20 = 11/20, which is 0.55. So that matches. Good.

Another check: let me pick a = 1. Original expression: (4 + 6*1)/5 - (1 + 3*1)/4 = (4 + 6)/5 - (1 + 3)/4 = 10/5 - 4/4 = 2 - 1 = 1. Simplified expression: (9*1 + 11)/20 = (9 + 11)/20 = 20/20 = 1. Perfect, that matches too.

Another check: a = 2. Original expression: (4 + 6*2)/5 - (1 + 3*2)/4 = (4 + 12)/5 - (1 + 6)/4 = 16/5 - 7/4. Let me compute that. 16/5 is 3.2, 7/4 is 1.75. 3.2 - 1.75 = 1.45. Simplified expression: (9*2 + 11)/20 = (18 + 11)/20 = 29/20 = 1.45. Correct again.

So seems like my answer is correct. Therefore, the expression as a single fraction is (9a + 11)/20. Alternatively, written as (11 + 9a)/20. Both are equivalent. Depending on the preferred order, but usually variables come first, so maybe 9a + 11 over 20.

Wait, but in the original problem, the first fraction is (4 + 6a)/5. So maybe they want the terms in the same order? But in the numerator, when we combined, it was 11 + 9a. But 9a + 11 is the same. So either way is acceptable.

Alternatively, maybe I can write it as (9a + 11)/20. Yeah, that's probably the standard way. So I think that's the answer.

Just to recap the steps:

1. Find common denominator: 20.
2. Convert each fraction:
   - (4 + 6a)/5 = (16 + 24a)/20
   - (1 + 3a)/4 = (5 + 15a)/20
3. Subtract the second fraction from the first:
   - (16 + 24a - 5 - 15a)/20
4. Combine like terms:
   - 16 - 5 = 11
   - 24a - 15a = 9a
5. Result: (11 + 9a)/20 or (9a + 11)/20.

All steps check out. So I think that's the correct answer.

**Final Answer**
The expression as a single fraction is \boxed{\dfrac{9a + 11}{20}}.

To simplify the expression
$$
\frac{4+6a}{5} - \frac{1+3a}{4}
$$
into a single fraction, we follow a systematic process of finding a common denominator and combining the numerators.

---

### **Step 1: Find the Least Common Denominator (LCD)**

The denominators are 5 and 4. Since they are relatively prime, their least common denominator is:

$$
\text{LCD} = 5 \times 4 = 20
$$

---

### **Step 2: Convert Each Fraction to Have the Common Denominator**

We rewrite each fraction with denominator 20:

- For $\frac{4 + 6a}{5}$, multiply numerator and denominator by 4:
  $$
  \frac{4 + 6a}{5} = \frac{4(4 + 6a)}{20} = \frac{16 + 24a}{20}
  $$

- For $\frac{1 + 3a}{4}$, multiply numerator and denominator by 5:
  $$
  \frac{1 + 3a}{4} = \frac{5(1 + 3a)}{20} = \frac{5 + 15a}{20}
  $$

---

### **Step 3: Subtract the Two Fractions**

Now that both fractions have the same denominator, we subtract the numerators:

$$
\frac{16 + 24a}{20} - \frac{5 + 15a}{20} = \frac{(16 + 24a) - (5 + 15a)}{20}
$$

Distribute the subtraction across the second numerator:

$$
= \frac{16 + 24a - 5 - 15a}{20}
$$

---

### **Step 4: Combine Like Terms**

Group and simplify the terms in the numerator:

- Constant terms: $16 - 5 = 11$
- Terms with $a$: $24a - 15a = 9a$

So the numerator becomes:

$$
11 + 9a
$$

---

### **Step 5: Write the Final Answer**

Now, the expression is a single fraction:

$$
\frac{11 + 9a}{20}
$$

This is the simplified form of the original expression.

---

### **Final Answer**

$$
\boxed{\dfrac{9a + 11}{20}}
$$

Solution: The common denominator of $5$ and $4$ is $20$, so we multiply top and bottom of the first fraction by $4$ and multiply top and bottom of the second fraction by $5$. We get \[\frac{4(4+6a)}{4 \cdot 5} - \frac{5(1+3a)}{4 \cdot 5} = \frac{16+24a}{20}-\frac{5+15a}{20}.\] We combine the fractions, being careful to put the numerator of the second fraction in parentheses (as we are subtracting the entire numerator), yielding \[\frac{16+24a-(5+15a)}{20} = \frac{16+24a-5-15a}{20}=\boxed{\frac{11+9a}{20}}.\]