Explanation:
Okay, so I need to find an integer x that satisfies the inequality 1/4 < x/7 < 1/3. Hmm, let me think about how to approach this.

First, maybe I should try to solve the inequality for x. Let me write down the inequality again:

1/4 < x/7 < 1/3.

I need to find the value of x that makes this true. Since all the terms are positive, I can multiply all parts of the inequality by 7 to eliminate the denominator. Let me do that step by step.

Starting with the left part of the inequality: 1/4 < x/7. If I multiply both sides by 7, that would give me (1/4)*7 < x. Let me calculate that. 7 divided by 4 is 1.75. So, 1.75 < x.

Now, the right part of the inequality: x/7 < 1/3. Multiplying both sides by 7 gives x < (1/3)*7. Calculating that, 7 divided by 3 is approximately 2.333... So, x < 2.333...

Putting both parts together, we have 1.75 < x < 2.333...

But x has to be an integer. Let me think. The integers between 1.75 and 2.333... Well, 1.75 is between 1 and 2, and 2.333... is between 2 and 3. So the only integer that lies strictly between 1.75 and 2.333 is 2. Because 2 is greater than 1.75 and less than 2.333...

Wait, let me check if that's correct. Let me verify by plugging x = 2 into the original inequality.

Original inequality: 1/4 < 2/7 < 1/3.

Calculating 1/4 is 0.25, 2/7 is approximately 0.2857, and 1/3 is approximately 0.3333. So 0.25 < 0.2857 < 0.3333. Yes, that's true. So x = 2 satisfies the inequality.

But let me make sure there are no other integers in that interval. The next integer above 2 is 3. Let me check x = 3. Then 3/7 is approximately 0.4286, which is greater than 1/3 (0.3333), so that doesn't work. The integer below 2 is 1. Let me check x = 1. Then 1/7 is approximately 0.1429, which is less than 1/4 (0.25), so that doesn't work either. Therefore, the only integer that satisfies the inequality is 2.

Wait, but let me think again. Maybe I should approach this by cross-multiplying or something else to confirm?

Alternatively, maybe I can convert the fractions to have the same denominator or compare them differently. Let me try that.

Original inequality: 1/4 < x/7 < 1/3.

Let me first compare 1/4 and 1/3. Since 1/4 is 0.25 and 1/3 is approximately 0.333..., so the middle value x/7 has to be between these two. So, if I can find x such that when divided by 7, it's between 0.25 and 0.333...

Alternatively, maybe I can write the inequalities as:

For the left inequality: 1/4 < x/7. Multiply both sides by 7: 7/4 < x. Which is 1.75 < x.

For the right inequality: x/7 < 1/3. Multiply both sides by 7: x < 7/3. Which is approximately 2.333...

So again, x must be greater than 1.75 and less than 2.333... So the only integer in that interval is 2. So x = 2.

Alternatively, maybe I can think of fractions. Let me think. If I have 1/4 < x/7 < 1/3, then cross-multiplying the first inequality:

1/4 < x/7. Cross-multiplying (since all numbers are positive, the inequality direction remains the same):

1*7 < 4*x => 7 < 4x => x > 7/4 => x > 1.75.

Similarly, for the second inequality: x/7 < 1/3. Cross-multiplying:

3x < 7 => x < 7/3 ≈ 2.333...

So again, same result. Therefore, x must be between 1.75 and 2.333..., so integer x is 2.

Alternatively, maybe I can think of fractions with denominator 7. Let me see. Let me find fractions between 1/4 and 1/3 with denominator 7. So, 1/4 is equal to 7/28, and 1/3 is equal to 7/21? Wait, no. Wait, 1/4 is 7/28? Wait, 1/4 is 7/28? Wait, 1/4 is equal to 7/28? Wait, 1/4 is 7/28? Let me check. 1/4 is equal to 7/28? No, 1/4 is equal to 7/28? Wait, 1/4 is equal to 7/28? Wait, 1/4 is equal to 7/28? Let me compute 7 divided by 28. That's 0.25, which is 1/4. Yes, so 1/4 is 7/28. Similarly, 1/3 is equal to 7/21? Wait, 7 divided by 21 is 1/3. Yes. So, the inequality is 7/28 < x/7 < 7/21.

But maybe this isn't helpful. Alternatively, maybe I can write x/7 as a fraction with denominator 7. So, x/7 is just x over 7. So, if I want to find x such that 1/4 < x/7 < 1/3, then maybe I can find x such that when you divide by 7, it's between those two fractions.

Alternatively, maybe I can find a common denominator for 1/4 and 1/3. The least common denominator is 12. So, 1/4 is 3/12 and 1/3 is 4/12. Therefore, the inequality is 3/12 < x/7 < 4/12. But I don't know if that helps directly. Maybe I can cross-multiply?

Alternatively, maybe I can think of x as being between 7*(1/4) and 7*(1/3). Which is exactly what I did earlier. So, 7*(1/4) is 7/4 = 1.75 and 7*(1/3) is 7/3 ≈ 2.333...

So, again, x must be between 1.75 and 2.333..., so integer x is 2.

Alternatively, maybe I can think of fractions between 1/4 and 1/3. Let me think. For example, 1/4 is 0.25, 1/3 is approximately 0.333. So, numbers between them. Let me think of fractions with small denominators. For example, 1/3 is approximately 0.333, so maybe 1/3 is 0.333... and 1/4 is 0.25. So, numbers between them. For example, 0.2857 is 2/7, which is approximately 0.2857. So, 2/7 is between 1/4 and 1/3. Therefore, x is 2.

Alternatively, maybe I can check if there are other fractions between 1/4 and 1/3 with denominator 7. Let me see. If I have denominator 7, then possible numerators are 1, 2, 3, 4, 5, 6. So, fractions are 1/7, 2/7, 3/7, 4/7, 5/7, 6/7. Let me convert them to decimals:

1/7 ≈ 0.1429

2/7 ≈ 0.2857

3/7 ≈ 0.4286

4/7 ≈ 0.5714

5/7 ≈ 0.7143

6/7 ≈ 0.8571

So, comparing these to 1/4 (0.25) and 1/3 (≈0.3333). So, 2/7 ≈ 0.2857 is between 0.25 and 0.3333. The next one is 3/7 ≈ 0.4286 which is higher than 1/3. So, only 2/7 is between 1/4 and 1/3. Therefore, x must be 2.

Therefore, the integer x is 2.

Wait, but let me check if there are other fractions between 1/4 and 1/3 with different denominators. But the question is about x/7, so the denominator is fixed as 7. Therefore, only x/7 is considered. Therefore, only 2/7 is between 1/4 and 1/3. Therefore, x=2 is the only integer.

Alternatively, maybe I can think of the inequality as:

Multiply all parts by 7: 7*(1/4) < x < 7*(1/3). Which is 7/4 < x < 7/3.

Calculating 7 divided by 4 is 1.75, and 7 divided by 3 is approximately 2.333... So, x must be greater than 1.75 and less than 2.333... So, the integers between 1.75 and 2.333... are only 2.

Therefore, x=2 is the only integer that satisfies the inequality.

I think that's thorough. Let me check once again. If x=2, then 2/7 ≈ 0.2857. 1/4 is 0.25, 1/3 ≈ 0.333. So yes, 0.25 < 0.2857 < 0.333. Correct.

If x=1, 1/7 ≈ 0.1429 < 0.25, so no. If x=3, 3/7 ≈ 0.4286 > 0.333, so no. Therefore, only x=2.

Therefore, the answer is 2.

**Final Answer**
The integer $ x $ is \boxed{2}.

To solve the inequality:

$$
\frac{1}{4} < \frac{x}{7} < \frac{1}{3}
$$

we aim to find an integer $ x $ that satisfies this condition.

---

### Step 1: Eliminate the Denominator

To simplify the inequality, we can multiply all parts by 7 (since 7 is positive, the inequality direction remains unchanged):

$$
7 \cdot \frac{1}{4} < x < 7 \cdot \frac{1}{3}
$$

$$
\frac{7}{4} < x < \frac{7}{3}
$$

Now compute the decimal values:

$$
\frac{7}{4} = 1.75 \quad \text{and} \quad \frac{7}{3} \approx 2.333\ldots
$$

So the inequality becomes:

$$
1.75 < x < 2.333\ldots
$$

---

### Step 2: Identify the Integer in the Interval

We are looking for an **integer** $ x $ that lies strictly between 1.75 and 2.333...

The only integer in this interval is:

$$
x = 2
$$

---

### Step 3: Verify the Solution

Let’s plug $ x = 2 $ back into the original inequality to confirm:

$$
\frac{1}{4} < \frac{2}{7} < \frac{1}{3}
$$

Convert to decimals for comparison:

- $ \frac{1}{4} = 0.25 $
- $ \frac{2}{7} \approx 0.2857 $
- $ \frac{1}{3} \approx 0.3333 $

So:

$$
0.25 < 0.2857 < 0.3333
$$

This confirms that $ x = 2 $ satisfies the inequality.

---

### Final Answer

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to find an integer x that satisfies the inequality 1/4 < x/7 < 1/3. Hmm, let me think about how to approach this.

First, maybe I should try to solve the inequality for x. Let me write down the inequality again:

1/4 < x/7 < 1/3.

I need to find the value of x that makes this true. Since all the terms are positive, I can multiply all parts of the inequality by 7 to eliminate the denominator. Let me do that step by step.

Starting with the left part of the inequality: 1/4 < x/7. If I multiply both sides by 7, that would give me (1/4)*7 < x. Let me calculate that. 7 divided by 4 is 1.75. So, 1.75 < x.

Now, the right part of the inequality: x/7 < 1/3. Multiplying both sides by 7 gives x < (1/3)*7. Calculating that, 7 divided by 3 is approximately 2.333... So, x < 2.333...

Putting both parts together, we have 1.75 < x < 2.333...

But x has to be an integer. Let me think. The integers between 1.75 and 2.333... Well, 1.75 is between 1 and 2, and 2.333... is between 2 and 3. So the only integer that lies strictly between 1.75 and 2.333 is 2. Because 2 is greater than 1.75 and less than 2.333...

Wait, let me check if that's correct. Let me verify by plugging x = 2 into the original inequality.

Original inequality: 1/4 < 2/7 < 1/3.

Calculating 1/4 is 0.25, 2/7 is approximately 0.2857, and 1/3 is approximately 0.3333. So 0.25 < 0.2857 < 0.3333. Yes, that's true. So x = 2 satisfies the inequality.

But let me make sure there are no other integers in that interval. The next integer above 2 is 3. Let me check x = 3. Then 3/7 is approximately 0.4286, which is greater than 1/3 (0.3333), so that doesn't work. The integer below 2 is 1. Let me check x = 1. Then 1/7 is approximately 0.1429, which is less than 1/4 (0.25), so that doesn't work either. Therefore, the only integer that satisfies the inequality is 2.

Wait, but let me think again. Maybe I should approach this by cross-multiplying or something else to confirm?

Alternatively, maybe I can convert the fractions to have the same denominator or compare them differently. Let me try that.

Original inequality: 1/4 < x/7 < 1/3.

Let me first compare 1/4 and 1/3. Since 1/4 is 0.25 and 1/3 is approximately 0.333..., so the middle value x/7 has to be between these two. So, if I can find x such that when divided by 7, it's between 0.25 and 0.333...

Alternatively, maybe I can write the inequalities as:

For the left inequality: 1/4 < x/7. Multiply both sides by 7: 7/4 < x. Which is 1.75 < x.

For the right inequality: x/7 < 1/3. Multiply both sides by 7: x < 7/3. Which is approximately 2.333...

So again, x must be greater than 1.75 and less than 2.333... So the only integer in that interval is 2. So x = 2.

Alternatively, maybe I can think of fractions. Let me think. If I have 1/4 < x/7 < 1/3, then cross-multiplying the first inequality:

1/4 < x/7. Cross-multiplying (since all numbers are positive, the inequality direction remains the same):

1*7 < 4*x => 7 < 4x => x > 7/4 => x > 1.75.

Similarly, for the second inequality: x/7 < 1/3. Cross-multiplying:

3x < 7 => x < 7/3 ≈ 2.333...

So again, same result. Therefore, x must be between 1.75 and 2.333..., so integer x is 2.

Alternatively, maybe I can think of fractions with denominator 7. Let me see. Let me find fractions between 1/4 and 1/3 with denominator 7. So, 1/4 is equal to 7/28, and 1/3 is equal to 7/21? Wait, no. Wait, 1/4 is 7/28? Wait, 1/4 is 7/28? Wait, 1/4 is equal to 7/28? Wait, 1/4 is 7/28? Let me check. 1/4 is equal to 7/28? No, 1/4 is equal to 7/28? Wait, 1/4 is equal to 7/28? Wait, 1/4 is equal to 7/28? Let me compute 7 divided by 28. That's 0.25, which is 1/4. Yes, so 1/4 is 7/28. Similarly, 1/3 is equal to 7/21? Wait, 7 divided by 21 is 1/3. Yes. So, the inequality is 7/28 < x/7 < 7/21.

But maybe this isn't helpful. Alternatively, maybe I can write x/7 as a fraction with denominator 7. So, x/7 is just x over 7. So, if I want to find x such that 1/4 < x/7 < 1/3, then maybe I can find x such that when you divide by 7, it's between those two fractions.

Alternatively, maybe I can find a common denominator for 1/4 and 1/3. The least common denominator is 12. So, 1/4 is 3/12 and 1/3 is 4/12. Therefore, the inequality is 3/12 < x/7 < 4/12. But I don't know if that helps directly. Maybe I can cross-multiply?

Alternatively, maybe I can think of x as being between 7*(1/4) and 7*(1/3). Which is exactly what I did earlier. So, 7*(1/4) is 7/4 = 1.75 and 7*(1/3) is 7/3 ≈ 2.333...

So, again, x must be between 1.75 and 2.333..., so integer x is 2.

Alternatively, maybe I can think of fractions between 1/4 and 1/3. Let me think. For example, 1/4 is 0.25, 1/3 is approximately 0.333. So, numbers between them. Let me think of fractions with small denominators. For example, 1/3 is approximately 0.333, so maybe 1/3 is 0.333... and 1/4 is 0.25. So, numbers between them. For example, 0.2857 is 2/7, which is approximately 0.2857. So, 2/7 is between 1/4 and 1/3. Therefore, x is 2.

Alternatively, maybe I can check if there are other fractions between 1/4 and 1/3 with denominator 7. Let me see. If I have denominator 7, then possible numerators are 1, 2, 3, 4, 5, 6. So, fractions are 1/7, 2/7, 3/7, 4/7, 5/7, 6/7. Let me convert them to decimals:

1/7 ≈ 0.1429

2/7 ≈ 0.2857

3/7 ≈ 0.4286

4/7 ≈ 0.5714

5/7 ≈ 0.7143

6/7 ≈ 0.8571

So, comparing these to 1/4 (0.25) and 1/3 (≈0.3333). So, 2/7 ≈ 0.2857 is between 0.25 and 0.3333. The next one is 3/7 ≈ 0.4286 which is higher than 1/3. So, only 2/7 is between 1/4 and 1/3. Therefore, x must be 2.

Therefore, the integer x is 2.

Wait, but let me check if there are other fractions between 1/4 and 1/3 with different denominators. But the question is about x/7, so the denominator is fixed as 7. Therefore, only x/7 is considered. Therefore, only 2/7 is between 1/4 and 1/3. Therefore, x=2 is the only integer.

Alternatively, maybe I can think of the inequality as:

Multiply all parts by 7: 7*(1/4) < x < 7*(1/3). Which is 7/4 < x < 7/3.

Calculating 7 divided by 4 is 1.75, and 7 divided by 3 is approximately 2.333... So, x must be greater than 1.75 and less than 2.333... So, the integers between 1.75 and 2.333... are only 2.

Therefore, x=2 is the only integer that satisfies the inequality.

I think that's thorough. Let me check once again. If x=2, then 2/7 ≈ 0.2857. 1/4 is 0.25, 1/3 ≈ 0.333. So yes, 0.25 < 0.2857 < 0.333. Correct.

If x=1, 1/7 ≈ 0.1429 < 0.25, so no. If x=3, 3/7 ≈ 0.4286 > 0.333, so no. Therefore, only x=2.

Therefore, the answer is 2.

**Final Answer**
The integer $ x $ is \boxed{2}.

To solve the inequality:

$$
\frac{1}{4} < \frac{x}{7} < \frac{1}{3}
$$

we aim to find an integer $ x $ that satisfies this condition.

---

### Step 1: Eliminate the Denominator

To simplify the inequality, we can multiply all parts by 7 (since 7 is positive, the inequality direction remains unchanged):

$$
7 \cdot \frac{1}{4} < x < 7 \cdot \frac{1}{3}
$$

$$
\frac{7}{4} < x < \frac{7}{3}
$$

Now compute the decimal values:

$$
\frac{7}{4} = 1.75 \quad \text{and} \quad \frac{7}{3} \approx 2.333\ldots
$$

So the inequality becomes:

$$
1.75 < x < 2.333\ldots
$$

---

### Step 2: Identify the Integer in the Interval

We are looking for an **integer** $ x $ that lies strictly between 1.75 and 2.333...

The only integer in this interval is:

$$
x = 2
$$

---

### Step 3: Verify the Solution

Let’s plug $ x = 2 $ back into the original inequality to confirm:

$$
\frac{1}{4} < \frac{2}{7} < \frac{1}{3}
$$

Convert to decimals for comparison:

- $ \frac{1}{4} = 0.25 $
- $ \frac{2}{7} \approx 0.2857 $
- $ \frac{1}{3} \approx 0.3333 $

So:

$$
0.25 < 0.2857 < 0.3333
$$

This confirms that $ x = 2 $ satisfies the inequality.

---

### Final Answer

$$
\boxed{2}
$$

Solution: Multiplying all expressions in the inequalities by $7$, we have $\frac74 < x < \frac73$. Since $\frac 74$ is between $1$ and $2$, and $\frac 73$ is between $2$ and $3$, the only integer $x$ between these two fractions is $\boxed{2}$.


Explanation:
Okay, let me try to figure out this problem. So, the question is: If t(x) = 3 - g(x) and g(x) = √x, then what is t(g(16))? Hmm, let me break this down step by step.

First, I need to understand what each part means. There's a function t(x) which is defined in terms of another function g(x). Specifically, t(x) is equal to 3 minus g(x). And then they tell me that g(x) is the square root of x. So, maybe I need to substitute g(x) into t(x)? Let me write that down.

So, t(x) = 3 - g(x). Since g(x) is √x, then substituting that into t(x) would give me t(x) = 3 - √x. Wait, is that right? Let me check. If I replace g(x) with √x, then yes, t(x) becomes 3 minus the square root of x. Okay, that seems straightforward.

But the question isn't asking for t(x) in general; it's asking for t(g(16)). Hmm, so I need to find t evaluated at g(16). Let me parse that. So, first, compute g(16), and then plug that result into t(x). So, step one: find g(16).

Since g(x) is √x, then g(16) is √16. What's the square root of 16? Well, √16 is 4, because 4 times 4 is 16. So, g(16) equals 4. Got that part.

Now, the next step is to compute t(g(16)), which is t(4), because g(16) is 4. So, now I need to find t(4). But t(x) is 3 - g(x), so substituting x with 4, that would be t(4) = 3 - g(4).

Wait, so now I need to find g(4). Since g(x) is √x, then g(4) is √4, which is 2. Therefore, t(4) = 3 - 2 = 1. Therefore, t(g(16)) is 1? Let me check my steps again to make sure I didn't make a mistake.

First, compute g(16): √16 = 4. Then, plug that into t(x): t(4) = 3 - g(4). Then compute g(4): √4 = 2. Then subtract that from 3: 3 - 2 = 1. Seems straightforward.

Alternatively, maybe there's another way to approach this? Let me think. Since t(x) is defined as 3 - g(x), and g(x) is √x, then maybe I can write t(g(x)) as 3 - g(g(x))? Wait, but the question is t(g(16)), not t(g(x)). So, maybe that's not necessary. Let me see.

Alternatively, if I wanted to write t(g(x)) in terms of x, then t(g(x)) would be 3 - g(g(x)). Which would be 3 - √(g(x)) = 3 - √(√x) = 3 - x^(1/4). But that seems more complicated. However, maybe if I use this approach for x = 16, then t(g(16)) would be 3 - √(g(16)) = 3 - √(4) = 3 - 2 = 1. So, same answer. So that seems to confirm it.

Wait, so both methods give me 1. So, that seems to check out. Let me make sure I didn't confuse the order of operations or the function compositions.

Another way to think about it: t(g(16)) is the same as t evaluated at the output of g(16). Since g(16) is 4, then t(4) is 3 - g(4). And since g(4) is 2, then 3 - 2 is 1. Yep, that's consistent.

Is there a chance I messed up the square roots? Let me check again. √16 is 4, because 4 squared is 16. √4 is 2, because 2 squared is 4. So, that's correct.

Alternatively, could there be a negative square root? For example, sometimes square roots can have both positive and negative roots, but in the context of functions, unless specified otherwise, the square root function typically refers to the principal (non-negative) root. So, √16 is 4, not -4. Similarly, √4 is 2, not -2. So, that's correct.

Therefore, I think my answer is right. Let me just recap all steps once again:

1. Given t(x) = 3 - g(x)
2. Given g(x) = √x
3. Therefore, t(x) = 3 - √x
4. Need to find t(g(16))
5. Compute g(16) = √16 = 4
6. Then compute t(4) = 3 - √4 = 3 - 2 = 1
7. Therefore, the answer is 1.

Alternatively, if I didn't substitute t(x) first, but instead did it step by step:

t(g(16)) = 3 - g(g(16)) = 3 - g(4) = 3 - √4 = 3 - 2 = 1. Same result.

So, both methods lead to the same answer. Therefore, I think I can be confident that the answer is 1.

Wait, but let me check if there's any alternative interpretation of the problem. For example, could t(g(16)) be interpreted differently? Like, maybe someone might think that t(g(16)) is t multiplied by g(16)? But no, the notation t(g(16)) is standard function composition, meaning t evaluated at g(16). So, that's not the case.

Alternatively, could there be a typo in the problem? For example, if it was t(g(x)) instead of t(g(16)), but no, the problem specifically says t(g(16)).

Another thing to check: Did I misread the original functions? Let me check again.

Original problem: If t(x) = 3 - g(x) and g(x) = √x, then what is t(g(16))?

Yes, that's exactly what it says. So, t(x) is 3 minus g(x), and g(x) is square root of x. So, no issues there.

Alternatively, maybe someone might think that t(x) is 3 - g(x), so if you have t(g(16)), you need to substitute g(16) into t(x). Which is exactly what I did.

Alternatively, if someone thought that t(g(16)) is 3 - g(16), but that would be incorrect because t(x) is 3 - g(x), so t(g(16)) is 3 - g(g(16)), not 3 - g(16).

So, if someone mistakenly thought that t(g(16)) is 3 - g(16), they would compute 3 - √16 = 3 - 4 = -1, which is wrong. But since I followed the correct steps, I got 1.

Therefore, I think my answer is correct.

Just to recap once more:

- Compute inner function first: g(16) = √16 = 4
- Then compute t(4) = 3 - g(4) = 3 - √4 = 3 - 2 = 1

Yep, that's solid.

Alternatively, if I wanted to write out the entire expression step by step:

t(g(16)) = 3 - g(g(16))
= 3 - g(√16)
= 3 - g(4)
= 3 - √4
= 3 - 2
= 1

Yes, that's exactly the same. So, no matter how I approach it, the answer is 1.

Therefore, I think I can confidently say that the answer is 1.

**Final Answer**
\boxed{1}

To solve the problem, we are given two functions:

- $ t(x) = 3 - g(x) $
- $ g(x) = \sqrt{x} $

We are asked to find the value of $ t(g(16)) $.

---

### Step 1: Evaluate $ g(16) $

Since $ g(x) = \sqrt{x} $, we substitute $ x = 16 $:

$$
g(16) = \sqrt{16} = 4
$$

---

### Step 2: Use the result in $ t(x) $

Now that we know $ g(16) = 4 $, we substitute this into $ t(x) $:

$$
t(g(16)) = t(4)
$$

Recall that $ t(x) = 3 - g(x) $, so:

$$
t(4) = 3 - g(4)
$$

---

### Step 3: Evaluate $ g(4) $

$$
g(4) = \sqrt{4} = 2
$$

---

### Step 4: Final Calculation

$$
t(4) = 3 - 2 = 1
$$

---

### Final Answer

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, the question is: If t(x) = 3 - g(x) and g(x) = √x, then what is t(g(16))? Hmm, let me break this down step by step.

First, I need to understand what each part means. There's a function t(x) which is defined in terms of another function g(x). Specifically, t(x) is equal to 3 minus g(x). And then they tell me that g(x) is the square root of x. So, maybe I need to substitute g(x) into t(x)? Let me write that down.

So, t(x) = 3 - g(x). Since g(x) is √x, then substituting that into t(x) would give me t(x) = 3 - √x. Wait, is that right? Let me check. If I replace g(x) with √x, then yes, t(x) becomes 3 minus the square root of x. Okay, that seems straightforward.

But the question isn't asking for t(x) in general; it's asking for t(g(16)). Hmm, so I need to find t evaluated at g(16). Let me parse that. So, first, compute g(16), and then plug that result into t(x). So, step one: find g(16).

Since g(x) is √x, then g(16) is √16. What's the square root of 16? Well, √16 is 4, because 4 times 4 is 16. So, g(16) equals 4. Got that part.

Now, the next step is to compute t(g(16)), which is t(4), because g(16) is 4. So, now I need to find t(4). But t(x) is 3 - g(x), so substituting x with 4, that would be t(4) = 3 - g(4).

Wait, so now I need to find g(4). Since g(x) is √x, then g(4) is √4, which is 2. Therefore, t(4) = 3 - 2 = 1. Therefore, t(g(16)) is 1? Let me check my steps again to make sure I didn't make a mistake.

First, compute g(16): √16 = 4. Then, plug that into t(x): t(4) = 3 - g(4). Then compute g(4): √4 = 2. Then subtract that from 3: 3 - 2 = 1. Seems straightforward.

Alternatively, maybe there's another way to approach this? Let me think. Since t(x) is defined as 3 - g(x), and g(x) is √x, then maybe I can write t(g(x)) as 3 - g(g(x))? Wait, but the question is t(g(16)), not t(g(x)). So, maybe that's not necessary. Let me see.

Alternatively, if I wanted to write t(g(x)) in terms of x, then t(g(x)) would be 3 - g(g(x)). Which would be 3 - √(g(x)) = 3 - √(√x) = 3 - x^(1/4). But that seems more complicated. However, maybe if I use this approach for x = 16, then t(g(16)) would be 3 - √(g(16)) = 3 - √(4) = 3 - 2 = 1. So, same answer. So that seems to confirm it.

Wait, so both methods give me 1. So, that seems to check out. Let me make sure I didn't confuse the order of operations or the function compositions.

Another way to think about it: t(g(16)) is the same as t evaluated at the output of g(16). Since g(16) is 4, then t(4) is 3 - g(4). And since g(4) is 2, then 3 - 2 is 1. Yep, that's consistent.

Is there a chance I messed up the square roots? Let me check again. √16 is 4, because 4 squared is 16. √4 is 2, because 2 squared is 4. So, that's correct.

Alternatively, could there be a negative square root? For example, sometimes square roots can have both positive and negative roots, but in the context of functions, unless specified otherwise, the square root function typically refers to the principal (non-negative) root. So, √16 is 4, not -4. Similarly, √4 is 2, not -2. So, that's correct.

Therefore, I think my answer is right. Let me just recap all steps once again:

1. Given t(x) = 3 - g(x)
2. Given g(x) = √x
3. Therefore, t(x) = 3 - √x
4. Need to find t(g(16))
5. Compute g(16) = √16 = 4
6. Then compute t(4) = 3 - √4 = 3 - 2 = 1
7. Therefore, the answer is 1.

Alternatively, if I didn't substitute t(x) first, but instead did it step by step:

t(g(16)) = 3 - g(g(16)) = 3 - g(4) = 3 - √4 = 3 - 2 = 1. Same result.

So, both methods lead to the same answer. Therefore, I think I can be confident that the answer is 1.

Wait, but let me check if there's any alternative interpretation of the problem. For example, could t(g(16)) be interpreted differently? Like, maybe someone might think that t(g(16)) is t multiplied by g(16)? But no, the notation t(g(16)) is standard function composition, meaning t evaluated at g(16). So, that's not the case.

Alternatively, could there be a typo in the problem? For example, if it was t(g(x)) instead of t(g(16)), but no, the problem specifically says t(g(16)).

Another thing to check: Did I misread the original functions? Let me check again.

Original problem: If t(x) = 3 - g(x) and g(x) = √x, then what is t(g(16))?

Yes, that's exactly what it says. So, t(x) is 3 minus g(x), and g(x) is square root of x. So, no issues there.

Alternatively, maybe someone might think that t(x) is 3 - g(x), so if you have t(g(16)), you need to substitute g(16) into t(x). Which is exactly what I did.

Alternatively, if someone thought that t(g(16)) is 3 - g(16), but that would be incorrect because t(x) is 3 - g(x), so t(g(16)) is 3 - g(g(16)), not 3 - g(16).

So, if someone mistakenly thought that t(g(16)) is 3 - g(16), they would compute 3 - √16 = 3 - 4 = -1, which is wrong. But since I followed the correct steps, I got 1.

Therefore, I think my answer is correct.

Just to recap once more:

- Compute inner function first: g(16) = √16 = 4
- Then compute t(4) = 3 - g(4) = 3 - √4 = 3 - 2 = 1

Yep, that's solid.

Alternatively, if I wanted to write out the entire expression step by step:

t(g(16)) = 3 - g(g(16))
= 3 - g(√16)
= 3 - g(4)
= 3 - √4
= 3 - 2
= 1

Yes, that's exactly the same. So, no matter how I approach it, the answer is 1.

Therefore, I think I can confidently say that the answer is 1.

**Final Answer**
\boxed{1}

To solve the problem, we are given two functions:

- $ t(x) = 3 - g(x) $
- $ g(x) = \sqrt{x} $

We are asked to find the value of $ t(g(16)) $.

---

### Step 1: Evaluate $ g(16) $

Since $ g(x) = \sqrt{x} $, we substitute $ x = 16 $:

$$
g(16) = \sqrt{16} = 4
$$

---

### Step 2: Use the result in $ t(x) $

Now that we know $ g(16) = 4 $, we substitute this into $ t(x) $:

$$
t(g(16)) = t(4)
$$

Recall that $ t(x) = 3 - g(x) $, so:

$$
t(4) = 3 - g(4)
$$

---

### Step 3: Evaluate $ g(4) $

$$
g(4) = \sqrt{4} = 2
$$

---

### Step 4: Final Calculation

$$
t(4) = 3 - 2 = 1
$$

---

### Final Answer

$$
\boxed{1}
$$

Solution: We have $g(16) = 4$, so $t(g(16)) = t(4) = 3- g(4) = 3-\sqrt{4} = 3-2 = \boxed{1}$.


Explanation:
Okay, so I need to simplify this algebraic expression: 4 times (3r³ + 5r - 6) minus 6 times (2r³ - r² + 4r). And then express it in the form Ar² + Br + C, where A, B, and C are integers. Alright, let me take it step by step.

First, I remember that to simplify expressions like this, I need to distribute the coefficients outside the parentheses to each term inside. So, let me start with the first part: 4(3r³ + 5r - 6).

Let me distribute the 4 to each term inside the first set of parentheses.

So, 4 multiplied by 3r³ is... let me calculate that. 4*3 is 12, so that term becomes 12r³.

Next term is 4 multiplied by 5r. 4*5 is 20, so that term is 20r.

Then the last term is 4 multiplied by -6. 4*(-6) is -24.

So, after distributing the 4, the first part becomes 12r³ + 20r - 24.

Now, moving on to the second part of the expression: -6(2r³ - r² + 4r). Wait, there's a minus sign in front of the 6, so I need to distribute that as well.

Let me make sure I handle the signs correctly. The expression is minus 6 times the entire parentheses. So, I can think of it as -6*(2r³) + (-6)*(-r²) + (-6)*(4r).

Calculating each term:

First term: -6*2r³. That's -12r³.

Second term: -6*(-r²). Multiplying two negatives gives a positive. So, that's +6r².

Third term: -6*(4r). That's -24r.

So, after distributing the -6, the second part becomes -12r³ + 6r² - 24r.

Now, I need to combine the two parts: the first part (12r³ + 20r - 24) and the second part (-12r³ + 6r² - 24r).

Let me write them out together:

12r³ + 20r - 24 -12r³ + 6r² - 24r.

Now, I need to combine like terms. Let me list all the terms:

- The r³ terms: 12r³ and -12r³.

- The r² terms: There's a +6r².

- The r terms: 20r and -24r.

- The constant term: -24.

Let me handle each type of term one by one.

Starting with the r³ terms: 12r³ - 12r³. That should be 0. Because 12 minus 12 is 0. So, those terms cancel out.

Next, the r² term: only 6r². So that stays as is.

Then the r terms: 20r - 24r. Let me compute that. 20 - 24 is -4, so that term is -4r.

Then the constant term is -24.

So putting it all together, after combining like terms, the expression simplifies to:

6r² - 4r - 24.

Wait, but the problem says to express it in the form Ar² + Br + C. Let me check if I did everything correctly. Let me go through the steps again to make sure I didn't make a mistake.

First part: 4*(3r³ + 5r -6) = 12r³ + 20r -24. That seems right. 4*3r³ is 12r³, 4*5r is 20r, 4*(-6) is -24. Correct.

Second part: -6*(2r³ - r² +4r). Let me check that again. Distribute -6:

-6*2r³ = -12r³. Correct.

-6*(-r²) = +6r². Correct.

-6*(4r) = -24r. Correct. So that part is -12r³ +6r² -24r. Correct.

Then combining:

12r³ -12r³ is 0. Correct.

Then 6r². Then 20r -24r is -4r. Then -24. So the simplified expression is 6r² -4r -24.

But wait, the problem says to express it in the form Ar² + Br + C. So that would be A=6, B=-4, C=-24.

But let me check again if there are any other terms or if I missed something. Let me verify by expanding again.

Alternatively, maybe I can check by plugging in a value for r and see if both the original expression and the simplified one give the same result. Let me try r=1.

Original expression: 4*(3(1)^3 +5(1) -6) -6*(2(1)^3 - (1)^2 +4(1)).

Calculating inside the first parentheses: 3(1) +5(1) -6 = 3 +5 -6 = 2. Multiply by 4: 4*2=8.

Inside the second parentheses: 2(1) -1 +4(1) = 2 -1 +4 = 5. Multiply by -6: -6*5 = -30.

So total expression: 8 -30 = -22.

Now, simplified expression: 6(1)^2 -4(1) -24 = 6 -4 -24 = -22. Same result. Good.

Let me try another value, say r=2.

Original expression:

First part: 4*(3*(8) +5*(2) -6) = 4*(24 +10 -6) = 4*(28) = 112.

Second part: -6*(2*(8) - (4) +4*(2)) = -6*(16 -4 +8) = -6*(20) = -120.

Total expression: 112 -120 = -8.

Simplified expression: 6*(4) -4*(2) -24 = 24 -8 -24 = -8. Same result. Good.

Another check: r=0.

Original expression:

First part: 4*(0 +0 -6) = 4*(-6) = -24.

Second part: -6*(0 -0 +0) = -6*(0) = 0.

Total: -24 +0 = -24.

Simplified expression: 6*(0) -4*(0) -24 = -24. Correct.

So seems like my simplified expression is correct. Therefore, the answer is 6r² -4r -24.

But let me check if the problem requires the answer in the form Ar² + Br + C, where A, B, C are integers. In this case, A=6, B=-4, C=-24. All integers. So that's acceptable.

Wait, but maybe I need to write it as 6r² + (-4)r + (-24). But since they just want the form, it's okay to have negative coefficients.

Alternatively, maybe the problem expects the answer in a different form? Let me check again the original problem.

"Simplify 4(3r³+5r-6)-6(2r³-r²+4r), and express your answer in the form Ar² + Br + C, where A, B, and C are integers."

So, the simplified expression is 6r² -4r -24. Which is already in the form Ar² + Br + C. So that's the answer.

But let me check once more if I combined all terms correctly. Let me write down all terms again:

Original expansion:

First part: 12r³ +20r -24

Second part: -12r³ +6r² -24r

Combine:

12r³ -12r³ = 0

Then 6r²

Then 20r -24r = -4r

Then -24.

Yes, that's correct. So the answer is 6r² -4r -24.

Therefore, A is 6, B is -4, and C is -24.

I think that's correct. I don't see any mistakes in my calculations. The checks with specific values of r also confirm that the simplified expression is correct.

So, the final answer is 6r² -4r -24.

**Final Answer**
The simplified expression is \boxed{6r^2 - 4r - 24}.

To simplify the expression
$$
4(3r^3 + 5r - 6) - 6(2r^3 - r^2 + 4r),
$$
we follow a step-by-step process of **distributing** and **combining like terms**.

---

### Step 1: Distribute the coefficients

**First term:**
$$
4(3r^3 + 5r - 6) = 4 \cdot 3r^3 + 4 \cdot 5r + 4 \cdot (-6) = 12r^3 + 20r - 24
$$

**Second term:**
$$
-6(2r^3 - r^2 + 4r) = -6 \cdot 2r^3 + (-6) \cdot (-r^2) + (-6) \cdot 4r = -12r^3 + 6r^2 - 24r
$$

---

### Step 2: Combine the results

Now, combine the two expanded expressions:

$$
(12r^3 + 20r - 24) + (-12r^3 + 6r^2 - 24r)
$$

Group like terms:

- $ r^3 $: $ 12r^3 - 12r^3 = 0 $
- $ r^2 $: $ 6r^2 $
- $ r $: $ 20r - 24r = -4r $
- Constants: $ -24 $

---

### Step 3: Write the simplified expression

$$
6r^2 - 4r - 24
$$

This is now in the form $ Ar^2 + Br + C $, where:

- $ A = 6 $
- $ B = -4 $
- $ C = -24 $

---

### Final Answer

$$
\boxed{6r^2 - 4r - 24}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this algebraic expression: 4 times (3r³ + 5r - 6) minus 6 times (2r³ - r² + 4r). And then express it in the form Ar² + Br + C, where A, B, and C are integers. Alright, let me take it step by step.

First, I remember that to simplify expressions like this, I need to distribute the coefficients outside the parentheses to each term inside. So, let me start with the first part: 4(3r³ + 5r - 6).

Let me distribute the 4 to each term inside the first set of parentheses.

So, 4 multiplied by 3r³ is... let me calculate that. 4*3 is 12, so that term becomes 12r³.

Next term is 4 multiplied by 5r. 4*5 is 20, so that term is 20r.

Then the last term is 4 multiplied by -6. 4*(-6) is -24.

So, after distributing the 4, the first part becomes 12r³ + 20r - 24.

Now, moving on to the second part of the expression: -6(2r³ - r² + 4r). Wait, there's a minus sign in front of the 6, so I need to distribute that as well.

Let me make sure I handle the signs correctly. The expression is minus 6 times the entire parentheses. So, I can think of it as -6*(2r³) + (-6)*(-r²) + (-6)*(4r).

Calculating each term:

First term: -6*2r³. That's -12r³.

Second term: -6*(-r²). Multiplying two negatives gives a positive. So, that's +6r².

Third term: -6*(4r). That's -24r.

So, after distributing the -6, the second part becomes -12r³ + 6r² - 24r.

Now, I need to combine the two parts: the first part (12r³ + 20r - 24) and the second part (-12r³ + 6r² - 24r).

Let me write them out together:

12r³ + 20r - 24 -12r³ + 6r² - 24r.

Now, I need to combine like terms. Let me list all the terms:

- The r³ terms: 12r³ and -12r³.

- The r² terms: There's a +6r².

- The r terms: 20r and -24r.

- The constant term: -24.

Let me handle each type of term one by one.

Starting with the r³ terms: 12r³ - 12r³. That should be 0. Because 12 minus 12 is 0. So, those terms cancel out.

Next, the r² term: only 6r². So that stays as is.

Then the r terms: 20r - 24r. Let me compute that. 20 - 24 is -4, so that term is -4r.

Then the constant term is -24.

So putting it all together, after combining like terms, the expression simplifies to:

6r² - 4r - 24.

Wait, but the problem says to express it in the form Ar² + Br + C. Let me check if I did everything correctly. Let me go through the steps again to make sure I didn't make a mistake.

First part: 4*(3r³ + 5r -6) = 12r³ + 20r -24. That seems right. 4*3r³ is 12r³, 4*5r is 20r, 4*(-6) is -24. Correct.

Second part: -6*(2r³ - r² +4r). Let me check that again. Distribute -6:

-6*2r³ = -12r³. Correct.

-6*(-r²) = +6r². Correct.

-6*(4r) = -24r. Correct. So that part is -12r³ +6r² -24r. Correct.

Then combining:

12r³ -12r³ is 0. Correct.

Then 6r². Then 20r -24r is -4r. Then -24. So the simplified expression is 6r² -4r -24.

But wait, the problem says to express it in the form Ar² + Br + C. So that would be A=6, B=-4, C=-24.

But let me check again if there are any other terms or if I missed something. Let me verify by expanding again.

Alternatively, maybe I can check by plugging in a value for r and see if both the original expression and the simplified one give the same result. Let me try r=1.

Original expression: 4*(3(1)^3 +5(1) -6) -6*(2(1)^3 - (1)^2 +4(1)).

Calculating inside the first parentheses: 3(1) +5(1) -6 = 3 +5 -6 = 2. Multiply by 4: 4*2=8.

Inside the second parentheses: 2(1) -1 +4(1) = 2 -1 +4 = 5. Multiply by -6: -6*5 = -30.

So total expression: 8 -30 = -22.

Now, simplified expression: 6(1)^2 -4(1) -24 = 6 -4 -24 = -22. Same result. Good.

Let me try another value, say r=2.

Original expression:

First part: 4*(3*(8) +5*(2) -6) = 4*(24 +10 -6) = 4*(28) = 112.

Second part: -6*(2*(8) - (4) +4*(2)) = -6*(16 -4 +8) = -6*(20) = -120.

Total expression: 112 -120 = -8.

Simplified expression: 6*(4) -4*(2) -24 = 24 -8 -24 = -8. Same result. Good.

Another check: r=0.

Original expression:

First part: 4*(0 +0 -6) = 4*(-6) = -24.

Second part: -6*(0 -0 +0) = -6*(0) = 0.

Total: -24 +0 = -24.

Simplified expression: 6*(0) -4*(0) -24 = -24. Correct.

So seems like my simplified expression is correct. Therefore, the answer is 6r² -4r -24.

But let me check if the problem requires the answer in the form Ar² + Br + C, where A, B, C are integers. In this case, A=6, B=-4, C=-24. All integers. So that's acceptable.

Wait, but maybe I need to write it as 6r² + (-4)r + (-24). But since they just want the form, it's okay to have negative coefficients.

Alternatively, maybe the problem expects the answer in a different form? Let me check again the original problem.

"Simplify 4(3r³+5r-6)-6(2r³-r²+4r), and express your answer in the form Ar² + Br + C, where A, B, and C are integers."

So, the simplified expression is 6r² -4r -24. Which is already in the form Ar² + Br + C. So that's the answer.

But let me check once more if I combined all terms correctly. Let me write down all terms again:

Original expansion:

First part: 12r³ +20r -24

Second part: -12r³ +6r² -24r

Combine:

12r³ -12r³ = 0

Then 6r²

Then 20r -24r = -4r

Then -24.

Yes, that's correct. So the answer is 6r² -4r -24.

Therefore, A is 6, B is -4, and C is -24.

I think that's correct. I don't see any mistakes in my calculations. The checks with specific values of r also confirm that the simplified expression is correct.

So, the final answer is 6r² -4r -24.

**Final Answer**
The simplified expression is \boxed{6r^2 - 4r - 24}.

To simplify the expression
$$
4(3r^3 + 5r - 6) - 6(2r^3 - r^2 + 4r),
$$
we follow a step-by-step process of **distributing** and **combining like terms**.

---

### Step 1: Distribute the coefficients

**First term:**
$$
4(3r^3 + 5r - 6) = 4 \cdot 3r^3 + 4 \cdot 5r + 4 \cdot (-6) = 12r^3 + 20r - 24
$$

**Second term:**
$$
-6(2r^3 - r^2 + 4r) = -6 \cdot 2r^3 + (-6) \cdot (-r^2) + (-6) \cdot 4r = -12r^3 + 6r^2 - 24r
$$

---

### Step 2: Combine the results

Now, combine the two expanded expressions:

$$
(12r^3 + 20r - 24) + (-12r^3 + 6r^2 - 24r)
$$

Group like terms:

- $ r^3 $: $ 12r^3 - 12r^3 = 0 $
- $ r^2 $: $ 6r^2 $
- $ r $: $ 20r - 24r = -4r $
- Constants: $ -24 $

---

### Step 3: Write the simplified expression

$$
6r^2 - 4r - 24
$$

This is now in the form $ Ar^2 + Br + C $, where:

- $ A = 6 $
- $ B = -4 $
- $ C = -24 $

---

### Final Answer

$$
\boxed{6r^2 - 4r - 24}
$$

Solution: Using the distributive property and combining like terms, we have  $4(3r^3+5r-6)-6(2r^3-r^2+4r) = 12r^3+20r-24-12r^3+6r^2-24r.$ Simplifying, we get $\boxed{6r^2-4r-24}.$


Explanation:
Okay, so I need to simplify this product of cosines: cos(2π/15) * cos(4π/15) * cos(8π/15) * cos(16π/15). Hmm, let me think about how to approach this.

First, I remember that there are some trigonometric identities for products of cosines. Maybe I can use product-to-sum formulas? But with four terms, that might get complicated. Alternatively, I recall that sometimes products of cosines with angles that are multiples of each other can be simplified using recursive relations or by using complex numbers. Let me see.

Wait, the angles here are 2π/15, 4π/15, 8π/15, and 16π/15. Let me note that 16π/15 is the same as π + π/15, right? Because π is 15π/15, so adding π/15 gives 16π/15. And cosine of π + x is -cos(x). So cos(16π/15) = cos(π + π/15) = -cos(π/15). So maybe I can rewrite the original product as:

cos(2π/15) * cos(4π/15) * cos(8π/15) * (-cos(π/15)).

So that's -cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15).

Hmm, so now I have four terms: π/15, 2π/15, 4π/15, 8π/15. These angles seem to be doubling each time. Let me check: π/15, 2π/15, 4π/15, 8π/15. If I double 8π/15, I get 16π/15, which is the same as π + π/15, which we already considered. So maybe there's a pattern here.

I remember that there's a formula for the product of cosines where each angle is double the previous one. For example, there's a formula for cosθ * cos2θ * cos4θ * ... * cos(2^{n-1}θ). Maybe I can use that? Let me recall.

I think the formula is something like 2^{1 - n} * sin(2^n θ) / sinθ. Wait, let me check. Let me think of a simpler case. For example, if I have cosθ * cos2θ * cos4θ... up to n terms.

Let me try with n=1: cosθ. If I use the formula, it would be 2^{1 - 1} * sin(2^1 θ)/sinθ = 1 * sin(2θ)/sinθ = 2cosθ. Wait, that's not equal to cosθ. So maybe my formula is wrong.

Alternatively, maybe there's another identity. Let me think. There's an identity that relates the product of cosines with angles in geometric progression. Let me try to derive it.

Let me consider multiplying both sides by 2 sinθ. Let me take the product cosθ * cos2θ * cos4θ... Let me denote P = cosθ * cos2θ * cos4θ... Let me multiply P by 2 sinθ.

So 2 sinθ * P = 2 sinθ cosθ * cos2θ * cos4θ...

But 2 sinθ cosθ = sin2θ. Then, 2 sinθ * P = sin2θ * cos2θ * cos4θ...

Again, multiply by 2: 2 * 2 sinθ * P = 2 sin2θ cos2θ * cos4θ... = sin4θ * cos4θ...

Continuing this way, each time multiplying by 2, we get:

After n steps, 2^n sinθ * P = sin(2^n θ).

Therefore, P = sin(2^n θ) / (2^n sinθ).

So that formula is correct. So if we have the product of cosθ * cos2θ * cos4θ * ... * cos(2^{n-1}θ) = sin(2^n θ) / (2^n sinθ).

So in our case, let me check. Our angles are π/15, 2π/15, 4π/15, 8π/15. So starting with θ = π/15, then doubling each time. So that's four terms: θ, 2θ, 4θ, 8θ. So n=4? Wait, the formula is for n terms. Let me check:

If n=4, then the product is cosθ * cos2θ * cos4θ * cos8θ = sin(2^4 θ) / (2^4 sinθ) = sin(16θ)/(16 sinθ).

So in our case, θ = π/15. Therefore, the product would be sin(16π/15) / (16 sin(π/15)).

But wait, our product is cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15) = sin(16π/15)/(16 sin(π/15)).

But let me check if that's correct. Let me compute 2^4 θ = 16θ = 16*(π/15) = 16π/15. So yes.

So then, according to the formula, the product is sin(16π/15)/(16 sin(π/15)).

But then, in our original problem, we had the product as -cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15). So that would be - [sin(16π/15)/(16 sin(π/15))].

So let me compute sin(16π/15). 16π/15 is π + π/15, so sin(π + π/15) = -sin(π/15). Therefore, sin(16π/15) = -sin(π/15).

Therefore, substituting back, we have:

- [ (-sin(π/15)) / (16 sin(π/15)) ] = - [ -1 / 16 ] = 1/16.

Wait, so the entire product simplifies to 1/16? Let me check again step by step to make sure I didn't make a mistake.

Original product: cos(2π/15) * cos(4π/15) * cos(8π/15) * cos(16π/15).

We noticed that cos(16π/15) = cos(π + π/15) = -cos(π/15). Therefore, the product becomes:

cos(2π/15) * cos(4π/15) * cos(8π/15) * (-cos(π/15)) = - [cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15)].

Then, using the formula for the product of cosθ * cos2θ * cos4θ * cos8θ where θ = π/15.

So according to the formula, that product is sin(16θ)/(16 sinθ) = sin(16π/15)/(16 sin(π/15)).

As sin(16π/15) = sin(π + π/15) = -sin(π/15). Therefore, substituting:

sin(16π/15)/(16 sin(π/15)) = (-sin(π/15))/(16 sin(π/15)) = -1/16.

Therefore, the original product is - [ -1/16 ] = 1/16.

So the answer is 1/16? Let me check if that makes sense.

Alternatively, maybe I can verify with approximate values. Let me compute each cosine term numerically and multiply them.

First, compute each angle in radians:

2π/15 ≈ 2*3.1416/15 ≈ 0.4189 radians

4π/15 ≈ 0.8378 radians

8π/15 ≈ 1.6755 radians

16π/15 ≈ 3.3510 radians

Compute cos(0.4189): cos(2π/15) ≈ cos(24°) ≈ 0.9135

cos(4π/15) ≈ cos(48°) ≈ 0.6691

cos(8π/15) ≈ cos(96°) ≈ -0.1045

cos(16π/15) ≈ cos(192°) ≈ cos(180° + 12°) = -cos(12°) ≈ -0.9781

So multiplying them all together:

0.9135 * 0.6691 * (-0.1045) * (-0.9781)

First multiply 0.9135 * 0.6691 ≈ 0.6115

Then multiply by -0.1045: 0.6115 * (-0.1045) ≈ -0.0639

Then multiply by -0.9781: -0.0639 * (-0.9781) ≈ 0.0625

Which is approximately 1/16, since 1/16 = 0.0625. So that checks out!

Therefore, the answer is 1/16.

But let me think again if there's another way to approach this problem, just to confirm.

Alternatively, maybe using complex numbers or roots of unity? Let me try that approach.

Consider the 15th roots of unity. The roots are e^(2πik/15) for k = 0, 1, ..., 14.

But how does that relate to the product of cosines?

Alternatively, since we have angles that are multiples of π/15, maybe we can relate them to sine or cosine products.

Alternatively, note that the product of cosines can be related to the real parts of complex exponentials. But that might be more complicated.

Alternatively, since we have angles doubling each time, maybe there's a recursive relation.

But since the numerical check worked and the formula seems to apply, I think the answer is indeed 1/16.

Wait, but let me check again the formula. The formula says that the product cosθ * cos2θ * cos4θ * ... * cos(2^{n-1}θ) = sin(2^n θ)/(2^n sinθ).

In our case, θ = π/15, and n=4, since we have four terms: θ, 2θ, 4θ, 8θ. So 2^4 θ = 16θ.

So the formula gives sin(16θ)/(2^4 sinθ) = sin(16π/15)/(16 sin(π/15)).

Which is equal to -sin(π/15)/(16 sin(π/15)) = -1/16.

Therefore, the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15) = -1/16.

But our original product was - [ that product ] = -(-1/16) = 1/16.

Yes, that seems correct.

Alternatively, maybe there's another identity. For example, using multiple-angle identities.

Alternatively, since 15 is 3*5, maybe there's some relation with the sine of π/15 or something. But I think the previous method is solid.

Another way to check: Let me compute the product step by step using identities.

Let me take the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15).

Let me pair them. For example, first multiply cos(π/15) * cos(8π/15), and then cos(2π/15) * cos(4π/15).

Using the identity cosA cosB = [cos(A+B) + cos(A-B)] / 2.

First pair: cos(π/15) * cos(8π/15) = [cos(9π/15) + cos(-7π/15)] / 2 = [cos(3π/5) + cos(7π/15)] / 2.

Similarly, second pair: cos(2π/15) * cos(4π/15) = [cos(6π/15) + cos(-2π/15)] / 2 = [cos(2π/5) + cos(2π/15)] / 2.

So the entire product becomes [ [cos(3π/5) + cos(7π/15)] / 2 ] * [ [cos(2π/5) + cos(2π/15)] / 2 ]

Which is 1/4 * [cos(3π/5) + cos(7π/15)] * [cos(2π/5) + cos(2π/15)]

This seems more complicated, but maybe we can compute these terms.

First, let me note that cos(3π/5) is cos(108°) ≈ -0.3090, cos(2π/5) is cos(72°) ≈ 0.3090, cos(7π/15) is cos(84°) ≈ 0.1045, and cos(2π/15) is cos(24°) ≈ 0.9135.

So substituting approximate values:

[ -0.3090 + 0.1045 ] * [ 0.3090 + 0.9135 ] ≈ (-0.2045) * (1.2225) ≈ -0.249. Then multiply by 1/4: -0.249 / 4 ≈ -0.06225, which is approximately -1/16 ≈ -0.0625. Close enough, considering rounding errors.

Therefore, this method also gives approximately -1/16 for the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15), so multiplying by -1 gives 1/16.

Therefore, this confirms the result again.

Alternatively, maybe there's a way to use the identity for sin(θ) = 2 sin(θ/2) cos(θ/2). But I think the previous methods are sufficient.

Therefore, I think the answer is 1/16.

But just to make sure, let me check another approach.

Let me recall that in general, the product of cos(π/2^{n}) * cos(π/2^{n-1}) * ... but not sure.

Alternatively, since 15 is 2^4 - 1, which is 16 - 1. Wait, 2^4 is 16, so 16 - 1 = 15. So maybe there's a connection here.

In general, for n, the product of cos(π/(2^{n} - 1)) * cos(2π/(2^{n} - 1)) * ... * cos(2^{n-1}π/(2^{n} - 1)) equals 1/2^{n}.

Wait, in our case, 2^{4} - 1 = 15, so n=4. Then the product would be 1/2^4 = 1/16. But our product is similar but with angles π/15, 2π/15,... up to 8π/15. However, note that 8π/15 is 2^{3}π/15, and 2^{4}π/15 is 16π/15, which is π + π/15.

But in the general formula, if we have angles π/(2^n - 1), 2π/(2^n - 1), ..., 2^{n-1}π/(2^n - 1), then the product is 1/2^n.

But in our case, 2^n - 1 = 15, so n=4. Therefore, the product of cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15) = 1/16? Wait, but earlier we found that this product is -1/16.

Wait, there's a discrepancy here. Which one is correct?

Wait, according to our previous calculation, the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15) = -1/16. But according to this general formula, it should be 1/16.

So which is correct?

Wait, maybe the general formula has some conditions. Let me check.

Let me check for n=1: 2^1 - 1 = 1. The product would be cos(π/1) = cos(π) = -1. According to the formula 1/2^1 = 1/2. Which is not equal. So the formula is not correct.

Alternatively, maybe the formula is different.

Alternatively, perhaps the formula is for angles that are π/(2^n + 1) or something else.

Alternatively, maybe the formula is for the product of cos(π/(2^{n} + 1)) * ...

Alternatively, perhaps I need to check another source.

But since we already have two methods giving us -1/16 for the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15), and the numerical calculation confirms that, then maybe the general formula is different.

Alternatively, maybe the formula is for the product of cos(π/(2^{n} - 1)) * cos(2π/(2^{n} - 1)) * ... * cos(2^{n-1}π/(2^{n} - 1)) equals 1/2^{n} if the angles are in a certain range.

But in our case, 2^{n}π/(2^{n} - 1) = 16π/15, which is greater than π/2. So maybe the formula is different.

Alternatively, perhaps the formula is for angles less than π/2.

Alternatively, maybe the formula is for the product of cos(π/(2^{n} - 1)) * cos(2π/(2^{n} - 1)) * ... * cos(2^{n-1}π/(2^{n} - 1)) equals 1/2^{n} if n is such that 2^{n}π/(2^{n} - 1) < π/2.

But for n=4, 2^{4}π/(2^{4} - 1) = 16π/15 ≈ 3.35 radians, which is more than π/2 ≈ 1.57. So that formula might not apply here.

Therefore, perhaps my initial formula is correct, and the general formula is different.

In any case, since we have two methods giving us -1/16 for the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15), and the original problem's product is - times that, leading to 1/16, and numerical calculation confirms that, I think the answer is indeed 1/16.

Therefore, I can be confident that the simplified value is 1/16.

**Final Answer**
\boxed{\dfrac{1}{16}}

To simplify the expression:

$$
\cos \left( \frac{2 \pi}{15} \right) \cos \left( \frac{4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac{16 \pi}{15} \right),
$$

we begin by observing the pattern in the angles. Each angle is double the previous one, and the sequence is:

$$
\frac{2\pi}{15}, \frac{4\pi}{15}, \frac{8\pi}{15}, \frac{16\pi}{15}.
$$

---

### Step 1: Use Trigonometric Identity for Cosine of a Sum

We note that:

$$
\cos\left(\frac{16\pi}{15}\right) = \cos\left(\pi + \frac{\pi}{15}\right) = -\cos\left(\frac{\pi}{15}\right),
$$

since $\cos(\pi + x) = -\cos(x)$. Substituting this into the original expression, we get:

$$
\cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right) \cos\left(\frac{16\pi}{15}\right) = -\cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right).
$$

---

### Step 2: Apply the Product of Cosines with Doubling Angles

There is a known identity for the product of cosines with angles in geometric progression:

$$
\cos\theta \cdot \cos(2\theta) \cdot \cos(4\theta) \cdots \cos(2^{n-1}\theta) = \frac{\sin(2^n \theta)}{2^n \sin\theta}.
$$

In our case, we have:

$$
\cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right),
$$

which corresponds to $n = 4$, $\theta = \frac{\pi}{15}$. Applying the identity:

$$
\cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right) = \frac{\sin\left(2^4 \cdot \frac{\pi}{15}\right)}{2^4 \sin\left(\frac{\pi}{15}\right)} = \frac{\sin\left(\frac{16\pi}{15}\right)}{16 \sin\left(\frac{\pi}{15}\right)}.
$$

Now, recall that:

$$
\sin\left(\frac{16\pi}{15}\right) = \sin\left(\pi + \frac{\pi}{15}\right) = -\sin\left(\frac{\pi}{15}\right).
$$

Substituting this back:

$$
\frac{\sin\left(\frac{16\pi}{15}\right)}{16 \sin\left(\frac{\pi}{15}\right)} = \frac{-\sin\left(\frac{\pi}{15}\right)}{16 \sin\left(\frac{\pi}{15}\right)} = -\frac{1}{16}.
$$

---

### Step 3: Final Simplification

Recall that the original expression was:

$$
-\left[\cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right)\right] = -\left(-\frac{1}{16}\right) = \frac{1}{16}.
$$

---

### Final Answer

$$
\boxed{\dfrac{1}{16}}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this product of cosines: cos(2π/15) * cos(4π/15) * cos(8π/15) * cos(16π/15). Hmm, let me think about how to approach this.

First, I remember that there are some trigonometric identities for products of cosines. Maybe I can use product-to-sum formulas? But with four terms, that might get complicated. Alternatively, I recall that sometimes products of cosines with angles that are multiples of each other can be simplified using recursive relations or by using complex numbers. Let me see.

Wait, the angles here are 2π/15, 4π/15, 8π/15, and 16π/15. Let me note that 16π/15 is the same as π + π/15, right? Because π is 15π/15, so adding π/15 gives 16π/15. And cosine of π + x is -cos(x). So cos(16π/15) = cos(π + π/15) = -cos(π/15). So maybe I can rewrite the original product as:

cos(2π/15) * cos(4π/15) * cos(8π/15) * (-cos(π/15)).

So that's -cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15).

Hmm, so now I have four terms: π/15, 2π/15, 4π/15, 8π/15. These angles seem to be doubling each time. Let me check: π/15, 2π/15, 4π/15, 8π/15. If I double 8π/15, I get 16π/15, which is the same as π + π/15, which we already considered. So maybe there's a pattern here.

I remember that there's a formula for the product of cosines where each angle is double the previous one. For example, there's a formula for cosθ * cos2θ * cos4θ * ... * cos(2^{n-1}θ). Maybe I can use that? Let me recall.

I think the formula is something like 2^{1 - n} * sin(2^n θ) / sinθ. Wait, let me check. Let me think of a simpler case. For example, if I have cosθ * cos2θ * cos4θ... up to n terms.

Let me try with n=1: cosθ. If I use the formula, it would be 2^{1 - 1} * sin(2^1 θ)/sinθ = 1 * sin(2θ)/sinθ = 2cosθ. Wait, that's not equal to cosθ. So maybe my formula is wrong.

Alternatively, maybe there's another identity. Let me think. There's an identity that relates the product of cosines with angles in geometric progression. Let me try to derive it.

Let me consider multiplying both sides by 2 sinθ. Let me take the product cosθ * cos2θ * cos4θ... Let me denote P = cosθ * cos2θ * cos4θ... Let me multiply P by 2 sinθ.

So 2 sinθ * P = 2 sinθ cosθ * cos2θ * cos4θ...

But 2 sinθ cosθ = sin2θ. Then, 2 sinθ * P = sin2θ * cos2θ * cos4θ...

Again, multiply by 2: 2 * 2 sinθ * P = 2 sin2θ cos2θ * cos4θ... = sin4θ * cos4θ...

Continuing this way, each time multiplying by 2, we get:

After n steps, 2^n sinθ * P = sin(2^n θ).

Therefore, P = sin(2^n θ) / (2^n sinθ).

So that formula is correct. So if we have the product of cosθ * cos2θ * cos4θ * ... * cos(2^{n-1}θ) = sin(2^n θ) / (2^n sinθ).

So in our case, let me check. Our angles are π/15, 2π/15, 4π/15, 8π/15. So starting with θ = π/15, then doubling each time. So that's four terms: θ, 2θ, 4θ, 8θ. So n=4? Wait, the formula is for n terms. Let me check:

If n=4, then the product is cosθ * cos2θ * cos4θ * cos8θ = sin(2^4 θ) / (2^4 sinθ) = sin(16θ)/(16 sinθ).

So in our case, θ = π/15. Therefore, the product would be sin(16π/15) / (16 sin(π/15)).

But wait, our product is cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15) = sin(16π/15)/(16 sin(π/15)).

But let me check if that's correct. Let me compute 2^4 θ = 16θ = 16*(π/15) = 16π/15. So yes.

So then, according to the formula, the product is sin(16π/15)/(16 sin(π/15)).

But then, in our original problem, we had the product as -cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15). So that would be - [sin(16π/15)/(16 sin(π/15))].

So let me compute sin(16π/15). 16π/15 is π + π/15, so sin(π + π/15) = -sin(π/15). Therefore, sin(16π/15) = -sin(π/15).

Therefore, substituting back, we have:

- [ (-sin(π/15)) / (16 sin(π/15)) ] = - [ -1 / 16 ] = 1/16.

Wait, so the entire product simplifies to 1/16? Let me check again step by step to make sure I didn't make a mistake.

Original product: cos(2π/15) * cos(4π/15) * cos(8π/15) * cos(16π/15).

We noticed that cos(16π/15) = cos(π + π/15) = -cos(π/15). Therefore, the product becomes:

cos(2π/15) * cos(4π/15) * cos(8π/15) * (-cos(π/15)) = - [cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15)].

Then, using the formula for the product of cosθ * cos2θ * cos4θ * cos8θ where θ = π/15.

So according to the formula, that product is sin(16θ)/(16 sinθ) = sin(16π/15)/(16 sin(π/15)).

As sin(16π/15) = sin(π + π/15) = -sin(π/15). Therefore, substituting:

sin(16π/15)/(16 sin(π/15)) = (-sin(π/15))/(16 sin(π/15)) = -1/16.

Therefore, the original product is - [ -1/16 ] = 1/16.

So the answer is 1/16? Let me check if that makes sense.

Alternatively, maybe I can verify with approximate values. Let me compute each cosine term numerically and multiply them.

First, compute each angle in radians:

2π/15 ≈ 2*3.1416/15 ≈ 0.4189 radians

4π/15 ≈ 0.8378 radians

8π/15 ≈ 1.6755 radians

16π/15 ≈ 3.3510 radians

Compute cos(0.4189): cos(2π/15) ≈ cos(24°) ≈ 0.9135

cos(4π/15) ≈ cos(48°) ≈ 0.6691

cos(8π/15) ≈ cos(96°) ≈ -0.1045

cos(16π/15) ≈ cos(192°) ≈ cos(180° + 12°) = -cos(12°) ≈ -0.9781

So multiplying them all together:

0.9135 * 0.6691 * (-0.1045) * (-0.9781)

First multiply 0.9135 * 0.6691 ≈ 0.6115

Then multiply by -0.1045: 0.6115 * (-0.1045) ≈ -0.0639

Then multiply by -0.9781: -0.0639 * (-0.9781) ≈ 0.0625

Which is approximately 1/16, since 1/16 = 0.0625. So that checks out!

Therefore, the answer is 1/16.

But let me think again if there's another way to approach this problem, just to confirm.

Alternatively, maybe using complex numbers or roots of unity? Let me try that approach.

Consider the 15th roots of unity. The roots are e^(2πik/15) for k = 0, 1, ..., 14.

But how does that relate to the product of cosines?

Alternatively, since we have angles that are multiples of π/15, maybe we can relate them to sine or cosine products.

Alternatively, note that the product of cosines can be related to the real parts of complex exponentials. But that might be more complicated.

Alternatively, since we have angles doubling each time, maybe there's a recursive relation.

But since the numerical check worked and the formula seems to apply, I think the answer is indeed 1/16.

Wait, but let me check again the formula. The formula says that the product cosθ * cos2θ * cos4θ * ... * cos(2^{n-1}θ) = sin(2^n θ)/(2^n sinθ).

In our case, θ = π/15, and n=4, since we have four terms: θ, 2θ, 4θ, 8θ. So 2^4 θ = 16θ.

So the formula gives sin(16θ)/(2^4 sinθ) = sin(16π/15)/(16 sin(π/15)).

Which is equal to -sin(π/15)/(16 sin(π/15)) = -1/16.

Therefore, the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15) = -1/16.

But our original product was - [ that product ] = -(-1/16) = 1/16.

Yes, that seems correct.

Alternatively, maybe there's another identity. For example, using multiple-angle identities.

Alternatively, since 15 is 3*5, maybe there's some relation with the sine of π/15 or something. But I think the previous method is solid.

Another way to check: Let me compute the product step by step using identities.

Let me take the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15).

Let me pair them. For example, first multiply cos(π/15) * cos(8π/15), and then cos(2π/15) * cos(4π/15).

Using the identity cosA cosB = [cos(A+B) + cos(A-B)] / 2.

First pair: cos(π/15) * cos(8π/15) = [cos(9π/15) + cos(-7π/15)] / 2 = [cos(3π/5) + cos(7π/15)] / 2.

Similarly, second pair: cos(2π/15) * cos(4π/15) = [cos(6π/15) + cos(-2π/15)] / 2 = [cos(2π/5) + cos(2π/15)] / 2.

So the entire product becomes [ [cos(3π/5) + cos(7π/15)] / 2 ] * [ [cos(2π/5) + cos(2π/15)] / 2 ]

Which is 1/4 * [cos(3π/5) + cos(7π/15)] * [cos(2π/5) + cos(2π/15)]

This seems more complicated, but maybe we can compute these terms.

First, let me note that cos(3π/5) is cos(108°) ≈ -0.3090, cos(2π/5) is cos(72°) ≈ 0.3090, cos(7π/15) is cos(84°) ≈ 0.1045, and cos(2π/15) is cos(24°) ≈ 0.9135.

So substituting approximate values:

[ -0.3090 + 0.1045 ] * [ 0.3090 + 0.9135 ] ≈ (-0.2045) * (1.2225) ≈ -0.249. Then multiply by 1/4: -0.249 / 4 ≈ -0.06225, which is approximately -1/16 ≈ -0.0625. Close enough, considering rounding errors.

Therefore, this method also gives approximately -1/16 for the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15), so multiplying by -1 gives 1/16.

Therefore, this confirms the result again.

Alternatively, maybe there's a way to use the identity for sin(θ) = 2 sin(θ/2) cos(θ/2). But I think the previous methods are sufficient.

Therefore, I think the answer is 1/16.

But just to make sure, let me check another approach.

Let me recall that in general, the product of cos(π/2^{n}) * cos(π/2^{n-1}) * ... but not sure.

Alternatively, since 15 is 2^4 - 1, which is 16 - 1. Wait, 2^4 is 16, so 16 - 1 = 15. So maybe there's a connection here.

In general, for n, the product of cos(π/(2^{n} - 1)) * cos(2π/(2^{n} - 1)) * ... * cos(2^{n-1}π/(2^{n} - 1)) equals 1/2^{n}.

Wait, in our case, 2^{4} - 1 = 15, so n=4. Then the product would be 1/2^4 = 1/16. But our product is similar but with angles π/15, 2π/15,... up to 8π/15. However, note that 8π/15 is 2^{3}π/15, and 2^{4}π/15 is 16π/15, which is π + π/15.

But in the general formula, if we have angles π/(2^n - 1), 2π/(2^n - 1), ..., 2^{n-1}π/(2^n - 1), then the product is 1/2^n.

But in our case, 2^n - 1 = 15, so n=4. Therefore, the product of cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15) = 1/16? Wait, but earlier we found that this product is -1/16.

Wait, there's a discrepancy here. Which one is correct?

Wait, according to our previous calculation, the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15) = -1/16. But according to this general formula, it should be 1/16.

So which is correct?

Wait, maybe the general formula has some conditions. Let me check.

Let me check for n=1: 2^1 - 1 = 1. The product would be cos(π/1) = cos(π) = -1. According to the formula 1/2^1 = 1/2. Which is not equal. So the formula is not correct.

Alternatively, maybe the formula is different.

Alternatively, perhaps the formula is for angles that are π/(2^n + 1) or something else.

Alternatively, maybe the formula is for the product of cos(π/(2^{n} + 1)) * ...

Alternatively, perhaps I need to check another source.

But since we already have two methods giving us -1/16 for the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15), and the numerical calculation confirms that, then maybe the general formula is different.

Alternatively, maybe the formula is for the product of cos(π/(2^{n} - 1)) * cos(2π/(2^{n} - 1)) * ... * cos(2^{n-1}π/(2^{n} - 1)) equals 1/2^{n} if the angles are in a certain range.

But in our case, 2^{n}π/(2^{n} - 1) = 16π/15, which is greater than π/2. So maybe the formula is different.

Alternatively, perhaps the formula is for angles less than π/2.

Alternatively, maybe the formula is for the product of cos(π/(2^{n} - 1)) * cos(2π/(2^{n} - 1)) * ... * cos(2^{n-1}π/(2^{n} - 1)) equals 1/2^{n} if n is such that 2^{n}π/(2^{n} - 1) < π/2.

But for n=4, 2^{4}π/(2^{4} - 1) = 16π/15 ≈ 3.35 radians, which is more than π/2 ≈ 1.57. So that formula might not apply here.

Therefore, perhaps my initial formula is correct, and the general formula is different.

In any case, since we have two methods giving us -1/16 for the product cos(π/15) * cos(2π/15) * cos(4π/15) * cos(8π/15), and the original problem's product is - times that, leading to 1/16, and numerical calculation confirms that, I think the answer is indeed 1/16.

Therefore, I can be confident that the simplified value is 1/16.

**Final Answer**
\boxed{\dfrac{1}{16}}

To simplify the expression:

$$
\cos \left( \frac{2 \pi}{15} \right) \cos \left( \frac{4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac{16 \pi}{15} \right),
$$

we begin by observing the pattern in the angles. Each angle is double the previous one, and the sequence is:

$$
\frac{2\pi}{15}, \frac{4\pi}{15}, \frac{8\pi}{15}, \frac{16\pi}{15}.
$$

---

### Step 1: Use Trigonometric Identity for Cosine of a Sum

We note that:

$$
\cos\left(\frac{16\pi}{15}\right) = \cos\left(\pi + \frac{\pi}{15}\right) = -\cos\left(\frac{\pi}{15}\right),
$$

since $\cos(\pi + x) = -\cos(x)$. Substituting this into the original expression, we get:

$$
\cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right) \cos\left(\frac{16\pi}{15}\right) = -\cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right).
$$

---

### Step 2: Apply the Product of Cosines with Doubling Angles

There is a known identity for the product of cosines with angles in geometric progression:

$$
\cos\theta \cdot \cos(2\theta) \cdot \cos(4\theta) \cdots \cos(2^{n-1}\theta) = \frac{\sin(2^n \theta)}{2^n \sin\theta}.
$$

In our case, we have:

$$
\cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right),
$$

which corresponds to $n = 4$, $\theta = \frac{\pi}{15}$. Applying the identity:

$$
\cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right) = \frac{\sin\left(2^4 \cdot \frac{\pi}{15}\right)}{2^4 \sin\left(\frac{\pi}{15}\right)} = \frac{\sin\left(\frac{16\pi}{15}\right)}{16 \sin\left(\frac{\pi}{15}\right)}.
$$

Now, recall that:

$$
\sin\left(\frac{16\pi}{15}\right) = \sin\left(\pi + \frac{\pi}{15}\right) = -\sin\left(\frac{\pi}{15}\right).
$$

Substituting this back:

$$
\frac{\sin\left(\frac{16\pi}{15}\right)}{16 \sin\left(\frac{\pi}{15}\right)} = \frac{-\sin\left(\frac{\pi}{15}\right)}{16 \sin\left(\frac{\pi}{15}\right)} = -\frac{1}{16}.
$$

---

### Step 3: Final Simplification

Recall that the original expression was:

$$
-\left[\cos\left(\frac{\pi}{15}\right) \cos\left(\frac{2\pi}{15}\right) \cos\left(\frac{4\pi}{15}\right) \cos\left(\frac{8\pi}{15}\right)\right] = -\left(-\frac{1}{16}\right) = \frac{1}{16}.
$$

---

### Final Answer

$$
\boxed{\dfrac{1}{16}}
$$

Solution:
Let $x = \cos \left( \frac{2 \pi}{15} \right) \cos \left (\frac {4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right).$  Then by repeated application of the double angle formula,
\begin{align*}
x \sin \left( \frac{2 \pi}{15} \right) &= \sin \left( \frac{2 \pi}{15} \right) \cos \left( \frac{2 \pi}{15} \right) \cos \left (\frac {4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right) \\
&= \frac{1}{2} \sin \left( \frac{4 \pi}{15} \right) \cos \left (\frac {4 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right) \\
&= \frac{1}{4} \sin \left (\frac {8 \pi}{15} \right) \cos \left( \frac{8 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right) \\
&= \frac{1}{8} \sin \left( \frac{16 \pi}{15} \right) \cos \left( \frac {16 \pi}{15} \right) \\
&= \frac{1}{16} \sin \left( \frac{32 \pi}{15} \right) \\
&= \frac{1}{16} \sin \left( \frac{2 \pi}{15} \right),
\end{align*}so $x = \boxed{\frac{1}{16}}.$


Explanation:
Okay, so I need to find the value of $ b $ in the direction vector $ \begin{pmatrix} -7 \\ b \end{pmatrix} $ for the given line in the Asymptote figure. Let me try to visualize the problem first.

The Asymptote code draws a coordinate system with axes from -8 to 8. Points A and B are given as (-5, 4) and (-1, 3) respectively. Then, there are points L and R, which are the intersections of the line AB with the vertical lines x = -8 and x = 8. The line LR is drawn in red, which is the line we're interested in. The question says that this line has a direction vector of the form $ \begin{pmatrix} -7 \\ b \end{pmatrix} $, and we need to find $ b $.

First, let me recall that the direction vector of a line can be found by taking two points on the line and subtracting their coordinates. Since the line LR is determined by points A and B, maybe I can find the direction vector from A to B and then relate it to the given direction vector?

Wait, but the line LR is the same as the line AB, right? Because L and R are points where the line AB intersects the vertical lines x = -8 and x = 8. So, the line LR is just the line AB extended to those vertical lines. Therefore, the direction vector of line LR should be the same as the direction vector of line AB.

But the problem says the direction vector is of the form $ \begin{pmatrix} -7 \\ b \end{pmatrix} $. So, if I can find the direction vector of AB, then I can compare it to this form and find $ b $.

Let me compute the direction vector of AB. Point A is (-5, 4) and point B is (-1, 3). The direction vector from A to B is $ \vec{AB} = B - A = (-1 - (-5), 3 - 4) = (-1 + 5, -1) = (4, -1) $.

Wait, so the direction vector is (4, -1). But the problem says the direction vector is (-7, b). Hmm, so these two vectors should be scalar multiples of each other because they are direction vectors of the same line.

So, if $ \begin{pmatrix} 4 \\ -1 \end{pmatrix} $ is a direction vector, then any scalar multiple of this vector is also a direction vector. Therefore, there exists some scalar $ k $ such that:

$ -7 = 4k $ and $ b = -1k $.

Let me solve for $ k $ first. From the x-component: $ -7 = 4k $, so $ k = -7/4 $. Then, substituting into the y-component: $ b = -1 * (-7/4) = 7/4 $.

Wait, so $ b = 7/4 $? Let me check if this makes sense.

Alternatively, maybe I need to check if the direction vector given in the problem is not necessarily the same as AB's direction vector, but maybe it's another direction vector for the same line. Since direction vectors can be scaled, so if the direction vector is (-7, b), then it should be parallel to (4, -1). Therefore, the ratio of the components should be the same.

So, the direction vector (-7, b) should be parallel to (4, -1). Therefore, the slope of the line should be the same.

The slope of AB is (change in y)/(change in x) = (-1)/4.

Alternatively, the slope of the direction vector (-7, b) is $ b / (-7) $. Therefore, setting them equal:

$ b / (-7) = -1 / 4 $

Solving for $ b $:

Multiply both sides by -7:

$ b = (-1 / 4) * (-7) = 7/4 $

Same result. So, $ b = 7/4 $.

But let me make sure that I didn't make a mistake here. Let me verify by actually finding the equation of the line AB and then finding the direction vector.

First, the line AB passes through points A(-5, 4) and B(-1, 3). Let me find its slope:

Slope $ m = (3 - 4)/(-1 - (-5)) = (-1)/(4) = -1/4 $.

So, the slope is -1/4. Therefore, the direction vector can be written as (change in x, change in y) = (4, -1) as before, or any scalar multiple.

Alternatively, if we want a direction vector with x-component -7, then since the slope is -1/4, the direction vector would have a rise over run of -1/4. So, if the run (change in x) is -7, then the rise (change in y) should be (-1/4)*(-7) = 7/4. Therefore, the direction vector is (-7, 7/4), so $ b = 7/4 $.

Therefore, this seems consistent.

But let me check if there's another way to approach this problem, maybe using the coordinates of points L and R? The Asymptote code defines L as the intersection of line AB with x = -8, and R as the intersection with x = 8. Then, the line LR is drawn. Maybe I need to find the coordinates of L and R, then compute the direction vector of LR?

Let me try that approach as a verification.

First, find the equation of line AB.

We have two points A(-5, 4) and B(-1, 3). The slope is -1/4 as before. So, the equation of the line can be written in point-slope form. Let me use point A:

$ y - 4 = (-1/4)(x - (-5)) $

Simplify:

$ y - 4 = (-1/4)(x + 5) $

Multiply out:

$ y = (-1/4)x - 5/4 + 4 $

Convert 4 to fourths:

$ y = (-1/4)x - 5/4 + 16/4 $

$ y = (-1/4)x + 11/4 $

So, the equation of line AB is $ y = (-1/4)x + 11/4 $.

Now, find point L, which is the intersection of this line with x = -8.

Substitute x = -8 into the equation:

$ y = (-1/4)(-8) + 11/4 = 2 + 11/4 = 8/4 + 11/4 = 19/4 $

Therefore, point L is (-8, 19/4).

Similarly, find point R, which is the intersection with x = 8:

$ y = (-1/4)(8) + 11/4 = -2 + 11/4 = -8/4 + 11/4 = 3/4 $

Therefore, point R is (8, 3/4).

Now, the line LR goes from (-8, 19/4) to (8, 3/4). Let me compute the direction vector of LR.

The direction vector is R - L = (8 - (-8), 3/4 - 19/4) = (16, -16/4) = (16, -4).

Wait, that's (16, -4). But the problem says the direction vector is of the form (-7, b). So, these two vectors should be scalar multiples. Let me check if (16, -4) is a scalar multiple of (-7, b).

Let me see. Let me write the direction vector from LR as (16, -4). Let me simplify this vector by dividing by 4: (4, -1). Which is the same direction vector as AB. So, that's consistent with earlier results.

But the problem says the direction vector is (-7, b). So, we need to find a scalar multiple of (4, -1) that has an x-component of -7.

So, let me set up the proportion. Let me suppose that the direction vector is k*(4, -1) = (4k, -k). We want 4k = -7. Therefore, k = -7/4. Then, the y-component is -k = -(-7/4) = 7/4. Therefore, b = 7/4.

So, same answer as before. Therefore, this confirms that b is 7/4.

Alternatively, since the direction vector from LR is (16, -4), which can be simplified to (4, -1), but if we want a direction vector with x-component -7, we can scale (4, -1) by -7/4, which gives (-7, 7/4). Therefore, b is 7/4.

Therefore, regardless of the approach, the answer seems to be 7/4.

But let me check once more. Maybe I made a mistake in calculating points L and R? Let me verify the coordinates again.

For point L: x = -8.

Equation of line AB: y = (-1/4)x + 11/4.

So, plugging x = -8:

y = (-1/4)(-8) + 11/4 = 2 + 11/4.

Convert 2 to fourths: 8/4 + 11/4 = 19/4. So, y = 19/4. So, point L is (-8, 19/4). Correct.

For point R: x = 8.

y = (-1/4)(8) + 11/4 = -2 + 11/4.

Convert -2 to fourths: -8/4 + 11/4 = 3/4. So, point R is (8, 3/4). Correct.

Then, direction vector from L to R is (8 - (-8), 3/4 - 19/4) = (16, -16/4) = (16, -4). Correct.

So, direction vector (16, -4). If we want a direction vector with x-component -7, then we need to find a scalar multiple.

Let me think of direction vectors as being any vector that points in the same direction. So, if we have direction vector (16, -4), then any scalar multiple is also a direction vector. So, if we want the x-component to be -7, then:

Let me denote the direction vector as (16k, -4k). We want 16k = -7. Therefore, k = -7/16. Then, the y-component is -4k = -4*(-7/16) = 28/16 = 7/4. Therefore, same result.

Therefore, regardless of the method, b is 7/4.

Alternatively, maybe the problem is expecting me to use the coordinates of L and R to compute the direction vector? Let me check.

The direction vector from L to R is (16, -4). But the problem says the direction vector is (-7, b). So, if we want to write this direction vector as a scalar multiple of (16, -4), then:

(-7, b) = k*(16, -4). Therefore:

-7 = 16k => k = -7/16

Then, b = -4k = -4*(-7/16) = 28/16 = 7/4.

Same answer.

Therefore, all methods lead to b = 7/4.

But let me check once again if there's another interpretation. The problem says "the following line is parameterized, so that its direction vector is of the form...". Maybe the line is not AB, but LR? But LR is the same line as AB.

Alternatively, maybe the parameterization is different? For example, if the line is parameterized with a direction vector of (-7, b), then perhaps we need to find b such that this direction vector is consistent with the line LR.

But since we already found that the direction vector of LR is (16, -4), which is equivalent to (-7, 7/4) when scaled appropriately, then b is 7/4.

Alternatively, maybe the problem is expecting me to use the slope?

The slope of the line is -1/4, as we found earlier. The direction vector is (-7, b), so the slope is b / (-7). Therefore:

b / (-7) = -1/4 => b = (-1/4)*(-7) = 7/4.

Same answer.

Therefore, I think I can be confident that the answer is 7/4.

But just to make sure, let me think if there's any chance that the direction vector is supposed to be from R to L instead of L to R? But direction vectors can be in either direction, so if we take R to L, the direction vector would be (-16, 4), which is also a scalar multiple of (4, -1). If we want a direction vector with x-component -7, then again, scaling (-16, 4) by some factor. Let me check:

Let me suppose direction vector is (-16k, 4k). We want -16k = -7 => k = 7/16. Then, b = 4k = 4*(7/16) = 28/16 = 7/4. Same result.

Therefore, regardless of direction, the value of b is 7/4.

Therefore, I think the answer is 7/4.

But just to check if there's any possible mistake in calculation. Let me recompute the slope.

Slope between A(-5, 4) and B(-1, 3):

Change in y: 3 - 4 = -1

Change in x: -1 - (-5) = -1 + 5 = 4

So slope is -1/4. Correct.

Therefore, direction vector (4, -1).

If we want direction vector (-7, b), then slope is b / (-7) = -1/4. Therefore, solving for b:

b / (-7) = -1/4 => b = (-1/4)*(-7) = 7/4.

Yes.

Alternatively, if I think of the direction vector as (-7, b), then the line's slope is b / (-7). Since the line's slope is -1/4, then:

b / (-7) = -1/4 => b = (-1/4)*(-7) = 7/4.

Same result.

Therefore, I think I can be confident that the answer is 7/4.

But just to check if the Asymptote code gives any clues. The Asymptote code defines points A(-5,4) and B(-1,3). Then, L is the intersection of line AB with x=-8 and y-axis? Wait, no. The Asymptote code says:

L = extension(A, B, (-8,0), (-8,1));

R = extension(A, B, (8,0), (8,1));

So, extension(A, B, (-8,0), (-8,1)) means the intersection of line AB with the line through (-8,0) and (-8,1). But (-8,0) to (-8,1) is the vertical line x=-8. Similarly for R. Therefore, L is the intersection of AB with x=-8, and R is the intersection with x=8. Which is what I calculated earlier.

Therefore, the line LR is the same as AB. Therefore, direction vector is same as AB.

Therefore, all steps are correct.

Therefore, the answer is $ \boxed{\dfrac{7}{4}} $.

**Final Answer**
The value of $ b $ is \boxed{\dfrac{7}{4}}.

To determine the value of $ b $ in the direction vector $ \begin{pmatrix} -7 \\ b \end{pmatrix} $, we analyze the geometry of the line described in the Asymptote diagram.

---

### Step 1: Understand the Line

The line in question is defined by the points $ A = (-5, 4) $ and $ B = (-1, 3) $. The line is extended to intersect the vertical lines $ x = -8 $ and $ x = 8 $, forming the segment $ LR $, which is the same line as $ AB $.

Thus, the direction vector of the line is the same as the vector from point $ A $ to point $ B $, or any scalar multiple of it.

---

### Step 2: Compute the Direction Vector of Line AB

The direction vector from $ A $ to $ B $ is:

$$
\vec{AB} = B - A = (-1 - (-5), 3 - 4) = (4, -1)
$$

So, the direction vector of the line is $ \begin{pmatrix} 4 \\ -1 \end{pmatrix} $.

---

### Step 3: Relate to the Given Direction Vector

We are told that the direction vector is of the form $ \begin{pmatrix} -7 \\ b \end{pmatrix} $. Since direction vectors of the same line are scalar multiples of each other, we can write:

$$
\begin{pmatrix} -7 \\ b \end{pmatrix} = k \cdot \begin{pmatrix} 4 \\ -1 \end{pmatrix}
$$

This gives us the system of equations:

$$
-7 = 4k \quad \text{and} \quad b = -k
$$

Solving for $ k $:

$$
k = \frac{-7}{4}
$$

Then:

$$
b = -k = -\left( \frac{-7}{4} \right) = \frac{7}{4}
$$

---

### Step 4: Verification via Slope

Alternatively, the slope of the line is:

$$
m = \frac{3 - 4}{-1 - (-5)} = \frac{-1}{4}
$$

The slope of the direction vector $ \begin{pmatrix} -7 \\ b \end{pmatrix} $ is:

$$
\frac{b}{-7}
$$

Setting this equal to the slope of the line:

$$
\frac{b}{-7} = -\frac{1}{4} \Rightarrow b = \left( -\frac{1}{4} \right)(-7) = \frac{7}{4}
$$

---

### Final Answer

$$
\boxed{\dfrac{7}{4}}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of $ b $ in the direction vector $ \begin{pmatrix} -7 \\ b \end{pmatrix} $ for the given line in the Asymptote figure. Let me try to visualize the problem first.

The Asymptote code draws a coordinate system with axes from -8 to 8. Points A and B are given as (-5, 4) and (-1, 3) respectively. Then, there are points L and R, which are the intersections of the line AB with the vertical lines x = -8 and x = 8. The line LR is drawn in red, which is the line we're interested in. The question says that this line has a direction vector of the form $ \begin{pmatrix} -7 \\ b \end{pmatrix} $, and we need to find $ b $.

First, let me recall that the direction vector of a line can be found by taking two points on the line and subtracting their coordinates. Since the line LR is determined by points A and B, maybe I can find the direction vector from A to B and then relate it to the given direction vector?

Wait, but the line LR is the same as the line AB, right? Because L and R are points where the line AB intersects the vertical lines x = -8 and x = 8. So, the line LR is just the line AB extended to those vertical lines. Therefore, the direction vector of line LR should be the same as the direction vector of line AB.

But the problem says the direction vector is of the form $ \begin{pmatrix} -7 \\ b \end{pmatrix} $. So, if I can find the direction vector of AB, then I can compare it to this form and find $ b $.

Let me compute the direction vector of AB. Point A is (-5, 4) and point B is (-1, 3). The direction vector from A to B is $ \vec{AB} = B - A = (-1 - (-5), 3 - 4) = (-1 + 5, -1) = (4, -1) $.

Wait, so the direction vector is (4, -1). But the problem says the direction vector is (-7, b). Hmm, so these two vectors should be scalar multiples of each other because they are direction vectors of the same line.

So, if $ \begin{pmatrix} 4 \\ -1 \end{pmatrix} $ is a direction vector, then any scalar multiple of this vector is also a direction vector. Therefore, there exists some scalar $ k $ such that:

$ -7 = 4k $ and $ b = -1k $.

Let me solve for $ k $ first. From the x-component: $ -7 = 4k $, so $ k = -7/4 $. Then, substituting into the y-component: $ b = -1 * (-7/4) = 7/4 $.

Wait, so $ b = 7/4 $? Let me check if this makes sense.

Alternatively, maybe I need to check if the direction vector given in the problem is not necessarily the same as AB's direction vector, but maybe it's another direction vector for the same line. Since direction vectors can be scaled, so if the direction vector is (-7, b), then it should be parallel to (4, -1). Therefore, the ratio of the components should be the same.

So, the direction vector (-7, b) should be parallel to (4, -1). Therefore, the slope of the line should be the same.

The slope of AB is (change in y)/(change in x) = (-1)/4.

Alternatively, the slope of the direction vector (-7, b) is $ b / (-7) $. Therefore, setting them equal:

$ b / (-7) = -1 / 4 $

Solving for $ b $:

Multiply both sides by -7:

$ b = (-1 / 4) * (-7) = 7/4 $

Same result. So, $ b = 7/4 $.

But let me make sure that I didn't make a mistake here. Let me verify by actually finding the equation of the line AB and then finding the direction vector.

First, the line AB passes through points A(-5, 4) and B(-1, 3). Let me find its slope:

Slope $ m = (3 - 4)/(-1 - (-5)) = (-1)/(4) = -1/4 $.

So, the slope is -1/4. Therefore, the direction vector can be written as (change in x, change in y) = (4, -1) as before, or any scalar multiple.

Alternatively, if we want a direction vector with x-component -7, then since the slope is -1/4, the direction vector would have a rise over run of -1/4. So, if the run (change in x) is -7, then the rise (change in y) should be (-1/4)*(-7) = 7/4. Therefore, the direction vector is (-7, 7/4), so $ b = 7/4 $.

Therefore, this seems consistent.

But let me check if there's another way to approach this problem, maybe using the coordinates of points L and R? The Asymptote code defines L as the intersection of line AB with x = -8, and R as the intersection with x = 8. Then, the line LR is drawn. Maybe I need to find the coordinates of L and R, then compute the direction vector of LR?

Let me try that approach as a verification.

First, find the equation of line AB.

We have two points A(-5, 4) and B(-1, 3). The slope is -1/4 as before. So, the equation of the line can be written in point-slope form. Let me use point A:

$ y - 4 = (-1/4)(x - (-5)) $

Simplify:

$ y - 4 = (-1/4)(x + 5) $

Multiply out:

$ y = (-1/4)x - 5/4 + 4 $

Convert 4 to fourths:

$ y = (-1/4)x - 5/4 + 16/4 $

$ y = (-1/4)x + 11/4 $

So, the equation of line AB is $ y = (-1/4)x + 11/4 $.

Now, find point L, which is the intersection of this line with x = -8.

Substitute x = -8 into the equation:

$ y = (-1/4)(-8) + 11/4 = 2 + 11/4 = 8/4 + 11/4 = 19/4 $

Therefore, point L is (-8, 19/4).

Similarly, find point R, which is the intersection with x = 8:

$ y = (-1/4)(8) + 11/4 = -2 + 11/4 = -8/4 + 11/4 = 3/4 $

Therefore, point R is (8, 3/4).

Now, the line LR goes from (-8, 19/4) to (8, 3/4). Let me compute the direction vector of LR.

The direction vector is R - L = (8 - (-8), 3/4 - 19/4) = (16, -16/4) = (16, -4).

Wait, that's (16, -4). But the problem says the direction vector is of the form (-7, b). So, these two vectors should be scalar multiples. Let me check if (16, -4) is a scalar multiple of (-7, b).

Let me see. Let me write the direction vector from LR as (16, -4). Let me simplify this vector by dividing by 4: (4, -1). Which is the same direction vector as AB. So, that's consistent with earlier results.

But the problem says the direction vector is (-7, b). So, we need to find a scalar multiple of (4, -1) that has an x-component of -7.

So, let me set up the proportion. Let me suppose that the direction vector is k*(4, -1) = (4k, -k). We want 4k = -7. Therefore, k = -7/4. Then, the y-component is -k = -(-7/4) = 7/4. Therefore, b = 7/4.

So, same answer as before. Therefore, this confirms that b is 7/4.

Alternatively, since the direction vector from LR is (16, -4), which can be simplified to (4, -1), but if we want a direction vector with x-component -7, we can scale (4, -1) by -7/4, which gives (-7, 7/4). Therefore, b is 7/4.

Therefore, regardless of the approach, the answer seems to be 7/4.

But let me check once more. Maybe I made a mistake in calculating points L and R? Let me verify the coordinates again.

For point L: x = -8.

Equation of line AB: y = (-1/4)x + 11/4.

So, plugging x = -8:

y = (-1/4)(-8) + 11/4 = 2 + 11/4.

Convert 2 to fourths: 8/4 + 11/4 = 19/4. So, y = 19/4. So, point L is (-8, 19/4). Correct.

For point R: x = 8.

y = (-1/4)(8) + 11/4 = -2 + 11/4.

Convert -2 to fourths: -8/4 + 11/4 = 3/4. So, point R is (8, 3/4). Correct.

Then, direction vector from L to R is (8 - (-8), 3/4 - 19/4) = (16, -16/4) = (16, -4). Correct.

So, direction vector (16, -4). If we want a direction vector with x-component -7, then we need to find a scalar multiple.

Let me think of direction vectors as being any vector that points in the same direction. So, if we have direction vector (16, -4), then any scalar multiple is also a direction vector. So, if we want the x-component to be -7, then:

Let me denote the direction vector as (16k, -4k). We want 16k = -7. Therefore, k = -7/16. Then, the y-component is -4k = -4*(-7/16) = 28/16 = 7/4. Therefore, same result.

Therefore, regardless of the method, b is 7/4.

Alternatively, maybe the problem is expecting me to use the coordinates of L and R to compute the direction vector? Let me check.

The direction vector from L to R is (16, -4). But the problem says the direction vector is (-7, b). So, if we want to write this direction vector as a scalar multiple of (16, -4), then:

(-7, b) = k*(16, -4). Therefore:

-7 = 16k => k = -7/16

Then, b = -4k = -4*(-7/16) = 28/16 = 7/4.

Same answer.

Therefore, all methods lead to b = 7/4.

But let me check once again if there's another interpretation. The problem says "the following line is parameterized, so that its direction vector is of the form...". Maybe the line is not AB, but LR? But LR is the same line as AB.

Alternatively, maybe the parameterization is different? For example, if the line is parameterized with a direction vector of (-7, b), then perhaps we need to find b such that this direction vector is consistent with the line LR.

But since we already found that the direction vector of LR is (16, -4), which is equivalent to (-7, 7/4) when scaled appropriately, then b is 7/4.

Alternatively, maybe the problem is expecting me to use the slope?

The slope of the line is -1/4, as we found earlier. The direction vector is (-7, b), so the slope is b / (-7). Therefore:

b / (-7) = -1/4 => b = (-1/4)*(-7) = 7/4.

Same answer.

Therefore, I think I can be confident that the answer is 7/4.

But just to make sure, let me think if there's any chance that the direction vector is supposed to be from R to L instead of L to R? But direction vectors can be in either direction, so if we take R to L, the direction vector would be (-16, 4), which is also a scalar multiple of (4, -1). If we want a direction vector with x-component -7, then again, scaling (-16, 4) by some factor. Let me check:

Let me suppose direction vector is (-16k, 4k). We want -16k = -7 => k = 7/16. Then, b = 4k = 4*(7/16) = 28/16 = 7/4. Same result.

Therefore, regardless of direction, the value of b is 7/4.

Therefore, I think the answer is 7/4.

But just to check if there's any possible mistake in calculation. Let me recompute the slope.

Slope between A(-5, 4) and B(-1, 3):

Change in y: 3 - 4 = -1

Change in x: -1 - (-5) = -1 + 5 = 4

So slope is -1/4. Correct.

Therefore, direction vector (4, -1).

If we want direction vector (-7, b), then slope is b / (-7) = -1/4. Therefore, solving for b:

b / (-7) = -1/4 => b = (-1/4)*(-7) = 7/4.

Yes.

Alternatively, if I think of the direction vector as (-7, b), then the line's slope is b / (-7). Since the line's slope is -1/4, then:

b / (-7) = -1/4 => b = (-1/4)*(-7) = 7/4.

Same result.

Therefore, I think I can be confident that the answer is 7/4.

But just to check if the Asymptote code gives any clues. The Asymptote code defines points A(-5,4) and B(-1,3). Then, L is the intersection of line AB with x=-8 and y-axis? Wait, no. The Asymptote code says:

L = extension(A, B, (-8,0), (-8,1));

R = extension(A, B, (8,0), (8,1));

So, extension(A, B, (-8,0), (-8,1)) means the intersection of line AB with the line through (-8,0) and (-8,1). But (-8,0) to (-8,1) is the vertical line x=-8. Similarly for R. Therefore, L is the intersection of AB with x=-8, and R is the intersection with x=8. Which is what I calculated earlier.

Therefore, the line LR is the same as AB. Therefore, direction vector is same as AB.

Therefore, all steps are correct.

Therefore, the answer is $ \boxed{\dfrac{7}{4}} $.

**Final Answer**
The value of $ b $ is \boxed{\dfrac{7}{4}}.

To determine the value of $ b $ in the direction vector $ \begin{pmatrix} -7 \\ b \end{pmatrix} $, we analyze the geometry of the line described in the Asymptote diagram.

---

### Step 1: Understand the Line

The line in question is defined by the points $ A = (-5, 4) $ and $ B = (-1, 3) $. The line is extended to intersect the vertical lines $ x = -8 $ and $ x = 8 $, forming the segment $ LR $, which is the same line as $ AB $.

Thus, the direction vector of the line is the same as the vector from point $ A $ to point $ B $, or any scalar multiple of it.

---

### Step 2: Compute the Direction Vector of Line AB

The direction vector from $ A $ to $ B $ is:

$$
\vec{AB} = B - A = (-1 - (-5), 3 - 4) = (4, -1)
$$

So, the direction vector of the line is $ \begin{pmatrix} 4 \\ -1 \end{pmatrix} $.

---

### Step 3: Relate to the Given Direction Vector

We are told that the direction vector is of the form $ \begin{pmatrix} -7 \\ b \end{pmatrix} $. Since direction vectors of the same line are scalar multiples of each other, we can write:

$$
\begin{pmatrix} -7 \\ b \end{pmatrix} = k \cdot \begin{pmatrix} 4 \\ -1 \end{pmatrix}
$$

This gives us the system of equations:

$$
-7 = 4k \quad \text{and} \quad b = -k
$$

Solving for $ k $:

$$
k = \frac{-7}{4}
$$

Then:

$$
b = -k = -\left( \frac{-7}{4} \right) = \frac{7}{4}
$$

---

### Step 4: Verification via Slope

Alternatively, the slope of the line is:

$$
m = \frac{3 - 4}{-1 - (-5)} = \frac{-1}{4}
$$

The slope of the direction vector $ \begin{pmatrix} -7 \\ b \end{pmatrix} $ is:

$$
\frac{b}{-7}
$$

Setting this equal to the slope of the line:

$$
\frac{b}{-7} = -\frac{1}{4} \Rightarrow b = \left( -\frac{1}{4} \right)(-7) = \frac{7}{4}
$$

---

### Final Answer

$$
\boxed{\dfrac{7}{4}}
$$

Solution:
The line passes through $\begin{pmatrix} -5 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 3 \end{pmatrix},$ so its direction vector is proportional to
\[\begin{pmatrix} -1 \\ 3 \end{pmatrix} - \begin{pmatrix} -5 \\ 4 \end{pmatrix} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}.\]To get an $x$-coordinate of $-7,$ we can multiply this vector by the scalar $-\frac{7}{4}.$  This gives us
\[-\frac{7}{4} \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} -7 \\ 7/4 \end{pmatrix}.\]Therefore, $b = \boxed{\frac{7}{4}}.$