Explanation:
Okay, so I need to figure out how many distinct values the function $ f(x) = \left\lfloor \left(-\frac{5}{8}\right)^x \right\rfloor $ can take when $ x $ is in the interval $[0, \infty)$. Alright, let me start by recalling what the floor function does. The floor function $ \lfloor y \rfloor $ gives the greatest integer less than or equal to $ y $. So, for any real number $ y $, $ \lfloor y \rfloor $ is the integer part of $ y $, essentially rounding down to the nearest integer.

Now, the function inside the floor is $ \left(-\frac{5}{8}\right)^x $. Let me think about how this behaves as $ x $ varies from 0 to infinity. First, let me note that $ -\frac{5}{8} $ is a negative number with absolute value less than 1. So, raising it to different powers will have different behaviors depending on whether the exponent is an integer or not. But since $ x $ is in $[0, \infty)$, it can be any non-negative real number, not just integers. Hmm, so I need to consider real exponents here.

Wait, but when dealing with exponents, if the base is negative, we have to be careful. For example, $ (-a)^x $ is only real if $ x $ is an integer or if $ a $ is positive and $ x $ is rational with an odd denominator, etc. But since $ x $ is a real number here, unless $ x $ is an integer, $ (-\frac{5}{8})^x $ might not be real. Wait, but the problem says that $ f(x) $ is defined for all $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number. So maybe the function is only defined for those $ x $ where $ (-\frac{5}{8})^x $ is real? But the problem says it's defined for all $ x $ in $[0, \infty)$. Hmm, maybe there's a misunderstanding here.

Wait, actually, if $ x $ is a real number, then $ (-\frac{5}{8})^x $ is not real unless $ x $ is an integer. Because if $ x $ is not an integer, then taking a negative number to a non-integer power would result in a complex number. However, the problem states that $ f(x) $ is defined for all $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number. So maybe the function is only defined for integer values of $ x $? But the problem says "defined for all values of $ x $ in $[0, \infty)$", which is confusing. Wait, maybe there's a mistake here?

Wait, let me check again. The problem says: "Let $ f(x) = \left\lfloor \left(-\frac{5}{8}\right)^x \right\rfloor $ be a function that is defined for all values of $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number." So, perhaps the function is only defined for those $ x $ where $ \left(-\frac{5}{8}\right)^x $ is real. But for real numbers, $ (-\frac{5}{8})^x $ is real only when $ x $ is an integer or when $ x $ is a rational number with an odd denominator? Wait, but even so, for real exponents, unless the base is positive, it's not real. So, for example, $ (-\frac{5}{8})^x $ is real only when $ x $ is an integer? Because otherwise, if $ x $ is not an integer, you can't take a negative number to a non-integer power and get a real number. Therefore, maybe the function is only defined for integer values of $ x $?

But the problem says "defined for all values of $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number." So maybe the domain is actually all real numbers $ x \geq 0 $, but only when $ (-\frac{5}{8})^x $ is real. However, since $ (-\frac{5}{8})^x $ is real only when $ x $ is an integer? Wait, but even for non-integer exponents, if you write it as $ e^{x \ln(-5/8)} $, but the logarithm of a negative number is not real. Therefore, $ (-\frac{5}{8})^x $ is not real for non-integer $ x $. Therefore, the function $ f(x) $ is only defined for integer values of $ x $ in $[0, \infty)$. Therefore, the domain is actually the non-negative integers. Therefore, the problem is actually about the function $ f(x) = \lfloor (-5/8)^x \rfloor $ where $ x $ is a non-negative integer. Therefore, the question is asking for the number of distinct values in the range of this function when $ x $ is a non-negative integer.

But the problem says "defined for all values of $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number." So maybe the problem is considering $ x $ as real numbers, but only when the expression is real. However, since for real $ x $, the expression is real only when $ x $ is an integer. Therefore, the domain is actually the non-negative integers. Therefore, the problem is equivalent to considering $ x $ as non-negative integers. Therefore, I need to evaluate $ f(x) = \lfloor (-5/8)^x \rfloor $ for $ x = 0, 1, 2, 3, ... $, and find how many distinct values there are.

Alternatively, maybe there's a different interpretation. Maybe the problem is considering complex numbers, but since the floor function is defined for real numbers, and the problem says $ f(x) $ is a real number, so the expression inside the floor must be real. Therefore, the domain is only those $ x $ for which $ (-5/8)^x $ is real. Which, as discussed, is only when $ x $ is an integer. Therefore, the domain is non-negative integers. Therefore, the problem reduces to considering integer exponents.

Therefore, proceeding with that assumption, let me consider $ x $ as non-negative integers. Let me compute $ (-5/8)^x $ for $ x = 0, 1, 2, 3, ... $, then take the floor of each, and see how many distinct values there are.

Starting with $ x = 0 $: $ (-5/8)^0 = 1 $. Therefore, $ f(0) = \lfloor 1 \rfloor = 1 $.

For $ x = 1 $: $ (-5/8)^1 = -5/8 = -0.625 $. The floor of -0.625 is -1, since -1 is the greatest integer less than or equal to -0.625. Therefore, $ f(1) = -1 $.

For $ x = 2 $: $ (-5/8)^2 = (25/64) ≈ 0.390625 $. The floor of that is 0. So $ f(2) = 0 $.

For $ x = 3 $: $ (-5/8)^3 = -125/512 ≈ -0.244140625 $. The floor of that is -1. So $ f(3) = -1 $.

For $ x = 4 $: $ (-5/8)^4 = 625/4096 ≈ 0.152587890625 $. Floor is 0. So $ f(4) = 0 $.

For $ x = 5 $: $ (-5/8)^5 = -3125/32768 ≈ -0.095367431640625 $. Floor is -1. So $ f(5) = -1 $.

For $ x = 6 $: $ (-5/8)^6 = 15625/262144 ≈ 0.059604644775390625 $. Floor is 0. So $ f(6) = 0 $.

Continuing this pattern, it seems that for even exponents, the value is positive and between 0 and 1, so floor is 0. For odd exponents, the value is negative and between -1 and 0, so floor is -1. However, let me check for higher exponents to see if this pattern continues.

For $ x = 7 $: $ (-5/8)^7 ≈ -0.07421875 $. Wait, no, let me compute it properly. Wait, $ (-5/8)^7 = - (5/8)^7. Let me compute $ (5/8)^7 $:

$ 5^7 = 78125 $, $ 8^7 = 2097152 $. So $ 78125 / 2097152 ≈ 0.03725290298... $. Therefore, $ (-5/8)^7 ≈ -0.0372529... $. So floor is -1. Similarly, for $ x = 8 $: $ (5/8)^8 ≈ 0.02685546875 $, so floor is 0. So yes, the pattern continues.

Therefore, for even $ x $, $ (-5/8)^x $ is positive and less than 1, so floor is 0. For odd $ x $, $ (-5/8)^x $ is negative and greater than -1, so floor is -1. However, wait, let me check for $ x = 0 $, which is even, and gives 1, which floor is 1. So, for $ x = 0 $, it's 1, and for all other even $ x \geq 2 $, it's between 0 and 1, so floor is 0. For odd $ x \geq 1 $, it's between -1 and 0, so floor is -1. Therefore, the function $ f(x) $ takes on three values: 1, 0, and -1. Wait, but let me check if there are any other values.

Wait, for $ x = 0 $, we have 1. For $ x = 1 $, -1. For $ x = 2 $, 0. For $ x = 3 $, -1. For $ x = 4 $, 0. Etc. So the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3? But let me check again.

Wait, but maybe there's a case where $ (-5/8)^x $ is exactly an integer? For example, when is $ (-5/8)^x $ an integer? For $ x = 0 $, it's 1, which is integer. For other $ x $, since $ 5/8 $ is a fraction, raising it to a positive integer power will give a fraction between 0 and 1 (for even exponents) or between -1 and 0 (for odd exponents). Therefore, the only integer value is 1 at $ x = 0 $. For all other $ x $, the value is between -1 and 1, but not including -1 or 1. Therefore, floor of those values would be 0 for positive values (since they are between 0 and 1) and -1 for negative values (since they are between -1 and 0). Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3?

But wait, let me check for $ x = 0 $, the value is 1, floor is 1. For $ x = 1 $, floor is -1. For $ x = 2 $, floor is 0. For $ x = 3 $, floor is -1. For $ x = 4 $, floor is 0. So yes, only three values. Therefore, the number of distinct values is 3. But let me check if there's any other possible value. For example, could $ (-5/8)^x $ ever be equal to 0? No, because $ (-5/8)^x $ approaches 0 as $ x $ approaches infinity, but never actually reaches 0. Similarly, it never reaches -1 or 1 except at $ x = 0 $ and $ x = 1 $? Wait, at $ x = 1 $, it's -5/8, which is -0.625, not -1. So floor is -1. So the only integer value is 1. Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3?

But wait, let me check for $ x = 0 $, which is allowed. So the function is defined for all $ x $ in $[0, \infty)$, but only when the expression is real. So if we take $ x $ as non-negative integers, then yes, the range is { -1, 0, 1 }, so three values. However, maybe there's a mistake here. Let me think again.

Wait, but maybe there's a case where $ (-5/8)^x $ is exactly an integer? For example, if $ x $ is such that $ (-5/8)^x $ is integer. Let me think. For $ x $ being a non-negative integer, $ (-5/8)^x $ is integer only when $ x = 0 $, because for $ x \geq 1 $, the denominator is 8^x, which is 2^{3x}, and the numerator is (-5)^x. So unless 8^x divides (-5)^x, which is only possible if x = 0. Therefore, only at x = 0, the value is integer. Therefore, floor of that is 1. For other x, the value is a fraction between -1 and 1 (excluding the endpoints), so floor is -1 or 0. Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3. But let me check again for x=0,1,2,3,...:

x=0: 1 → floor 1

x=1: -5/8 ≈ -0.625 → floor -1

x=2: 25/64 ≈ 0.390625 → floor 0

x=3: -125/512 ≈ -0.244 → floor -1

x=4: 625/4096 ≈ 0.1525 → floor 0

x=5: -3125/32768 ≈ -0.095 → floor -1

x=6: 15625/262144 ≈ 0.0596 → floor 0

x=7: -78125/2097152 ≈ -0.03725 → floor -1

x=8: 390625/16777216 ≈ 0.02328 → floor 0

So yes, the pattern continues. So the function takes values 1, -1, 0. Therefore, three distinct values. Therefore, the answer is 3. But wait, the problem says "the range of f(x)", so maybe there's something I'm missing here?

Wait, but let me check if there's any other possible value. For example, could the floor function ever give something else? For example, if $ (-5/8)^x $ is between 0 and 1, floor is 0. If it's between -1 and 0, floor is -1. If it's exactly 1, floor is 1. If it's exactly -1, floor is -1. But $ (-5/8)^x $ is never exactly -1 or 1 except at x=0. Because for x=0, it's 1. For x=1, it's -5/8, which is not -1. For x approaching infinity, $ (-5/8)^x $ approaches 0. So, the only integer value is 1. Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3. But let me check again.

Wait, but maybe there's a case where $ (-5/8)^x $ is equal to 0? But no, since $ (-5/8)^x $ approaches 0 as x approaches infinity, but never actually reaches 0. Therefore, floor of values approaching 0 from the positive side is 0, and from the negative side is -1. Therefore, the only integer value is 1. Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3. But let me check again for x=0, which is included in the domain. So yes, x=0 is allowed, and gives 1. So the answer is 3. Therefore, the number of distinct values is 3. Therefore, the answer is \boxed{3}?

Wait, but let me check again. Maybe there's a mistake in assuming that the domain is only integers. The problem says "defined for all values of x in [0, ∞) such that f(x) is a real number." So, if x is not an integer, then (-5/8)^x is not real. Therefore, the domain is only the non-negative integers. Therefore, the function is only defined at integer points. Therefore, the range is { -1, 0, 1 }, so three values. Therefore, answer is 3. But maybe the problem is considering x as real numbers, but somehow (-5/8)^x is real? Wait, but how?

Wait, another thought: if x is a rational number with an odd denominator, then (-5/8)^x can be real. For example, if x = 1/3, then (-5/8)^{1/3} is the cube root of -5/8, which is real. Similarly, x = 2/3 would be the cube root of (25/64), which is real. However, if x is irrational, then (-5/8)^x is not real. Therefore, if the problem allows x to be any real number where (-5/8)^x is real, then the domain would include more points. However, the problem says "defined for all values of x in [0, ∞) such that f(x) is a real number." So, the domain is all x in [0, ∞) for which (-5/8)^x is real. Therefore, the domain is all x in [0, ∞) where x is rational with odd denominator in reduced form, or x is integer. But this complicates things. However, the problem says "the function is defined for all values of x in [0, ∞) such that f(x) is a real number." So, maybe the problem is considering x as real numbers, but only when the expression is real. However, in that case, the function is defined on a dense set of points (like all rationals with odd denominators), but the problem is asking for the range. However, if x can be any real number where (-5/8)^x is real, then the function's range might be different. However, this seems more complicated, and the problem might be intended for integer exponents. But since the problem didn't specify that x is an integer, but said "defined for all values of x in [0, ∞) such that f(x) is a real number," maybe we need to consider real exponents where the expression is real.

But how can we handle that? Let me think. For real exponents, (-5/8)^x is real if and only if x is a rational number with an odd denominator when written in lowest terms, or x is an integer. However, since the problem says "defined for all values of x in [0, ∞)", but only when the expression is real. Therefore, the domain is the set of real numbers x ≥ 0 for which (-5/8)^x is real. However, this set is not straightforward. For example, for x irrational, it's not real. For x rational with even denominator, it's not real. For x rational with odd denominator, it's real. For x integer, it's real. Therefore, the domain is a countable dense subset of [0, ∞). However, the problem is asking for the range of f(x). If we consider all such x, then the function's range might be different. However, this seems complicated. But maybe the problem is intended for integer exponents. Since otherwise, the problem becomes more complex and the answer might not be straightforward.

Alternatively, maybe the problem is considering complex numbers, but since the floor function is defined for real numbers, and the problem says f(x) is a real number, so the expression inside must be real. Therefore, the domain is as discussed. However, if we consider x as real numbers where (-5/8)^x is real, then for x in [0, ∞), the expression is real only when x is an integer or x is a rational number with odd denominator. However, even so, the function's range would depend on the values of x. However, this seems too complicated for a problem that is likely expecting an answer of 3. Therefore, maybe the problem is intended for integer exponents. Therefore, proceeding with that assumption, the answer is 3. However, let me check again.

Alternatively, maybe the problem is considering x as real numbers, and interpreting (-5/8)^x as e^{x ln(-5/8)}, but since ln(-5/8) is not real, this is not possible. Therefore, the only way for (-5/8)^x to be real is if x is an integer. Therefore, the domain is integers. Therefore, the answer is 3. Therefore, I think the answer is 3. However, let me check once more.

Wait, but if x is allowed to be any real number where (-5/8)^x is real, then for example, x = 1/3, which is allowed, then (-5/8)^{1/3} is real. Let me compute that. The cube root of -5/8 is - (5/8)^{1/3} ≈ -0.855. Therefore, floor of that is -1. Similarly, for x = 2/3, (-5/8)^{2/3} is the cube root of (25/64) ≈ 0.7937, floor is 0. For x = 1/2, which is not allowed because denominator is even, so (-5/8)^{1/2} is not real. For x = 3/2, which is not allowed because denominator is even. Wait, x = 3/2 is 1.5, which is not an integer, and denominator is 2, which is even. Therefore, (-5/8)^{3/2} is not real. Therefore, only for x with odd denominators in reduced form. For example, x = 1/3, 2/3, 1/5, 2/5, etc. So, for such x, the value of (-5/8)^x is real. Therefore, the domain is more than just integers. Therefore, the function is defined on more points. Therefore, the range might include more values. Therefore, maybe my previous conclusion of 3 is incorrect.

Therefore, I need to consider this possibility. Let me think. If x is allowed to be any real number where (-5/8)^x is real, then for x in [0, ∞), the expression is real when x is an integer or x is a rational number with odd denominator. However, even so, the function's range might have more values. For example, take x = 1/3, which gives (-5/8)^{1/3} ≈ -0.855, floor is -1. Similarly, x = 2/3 gives ≈ 0.7937, floor is 0. x = 1/5: (-5/8)^{1/5} is the fifth root of -5/8 ≈ -0.86, floor is -1. x = 2/5: fifth root of (25/64) ≈ 0.83, floor is 0. Similarly, for x = 3/5: fifth root of (-5/8)^3 = fifth root of -125/512 ≈ -0.62, floor is -1. x = 4/5: fifth root of (5/8)^4 ≈ 0.59, floor is 0. So, similar to integer exponents, for non-integer rational exponents with odd denominators, the value is between -1 and 0 or 0 and 1, so floor is -1 or 0. Except for x=0, which gives 1. Therefore, even if we consider these rational exponents, the floor function still only gives -1, 0, or 1. Therefore, the range is still { -1, 0, 1 }, so three values. Therefore, even if we consider more points in the domain, the range doesn't increase. Therefore, the answer is still 3.

But wait, let me check for some other x. For example, take x approaching 0 from the right. Let me consider x approaching 0. For example, x = 0.1. But x = 0.1 is 1/10, which has denominator 10, even. Therefore, (-5/8)^{0.1} is not real. Therefore, not allowed. How about x = 1/1, which is integer. x = 1/3, which is allowed. What about x = 1/1, which is 1. So, for x approaching 0 from the right, but only for x where it's allowed. For example, x approaching 0 from the right through rational numbers with odd denominators. For example, x = 1/n where n is odd. Let me take n approaching infinity, so x approaching 0. Then (-5/8)^{1/n} approaches 1, since any number to the power 0 is 1. Therefore, for x approaching 0, the value approaches 1. Therefore, floor of that approaches 1. However, for x = 0, it's exactly 1. For x approaching 0 from the right, the value is approaching 1 from below or above? Let me check for x = 1/n, n odd. For example, n = 1: x = 1, value is -5/8 ≈ -0.625. For n = 3: x = 1/3, value ≈ -0.855. For n = 5: x = 1/5, value ≈ -0.86. Wait, as n increases, x approaches 0, and (-5/8)^{1/n} approaches 1? Wait, no. Wait, (-5/8)^{1/n} is equal to e^{(ln(-5/8))/n}. But ln(-5/8) is not real. Wait, but if we take x = 1/n with n odd, then (-5/8)^{1/n} is real. Let me compute it as (-5/8)^{1/n} = - (5/8)^{1/n} because n is odd. Therefore, for x = 1/n, the value is - (5/8)^{1/n}. As n approaches infinity, (5/8)^{1/n} approaches 1, so the value approaches -1. Therefore, for x approaching 0 from the right through odd denominators, the value approaches -1. Therefore, floor of that approaches -1. However, at x = 0, the value is 1. Therefore, there is a jump from 1 at x=0 to approaching -1 as x approaches 0 from the right. Therefore, the floor function at x=0 is 1, and for x approaching 0 from the right, floor is -1. Therefore, the value 1 is only achieved at x=0. For other x, the floor is either -1 or 0. Therefore, the range is still { -1, 0, 1 }, so three values. Therefore, even considering non-integer x where the expression is real, the range doesn't increase. Therefore, the answer is 3.

But let me check for some other x. For example, take x = 2/3. Then (-5/8)^{2/3} is equal to [(-5/8)^{1/3}]^2 = [ - (5/8)^{1/3} ]^2 = (5/8)^{2/3} ≈ (0.855)^2 ≈ 0.731, floor is 0. Similarly, for x = 3/3 = 1, floor is -1. For x = 4/3, which is 1.333..., which is allowed? Wait, x = 4/3. Let me check if it's allowed. 4/3 is a rational number with denominator 3, which is odd. Therefore, yes, allowed. Then (-5/8)^{4/3} = [(-5/8)^{1/3}]^4 = [ - (5/8)^{1/3} ]^4 = (5/8)^{4/3} ≈ (0.855)^4 ≈ 0.53, floor is 0. Similarly, for x = 5/3, which is 1.666..., allowed. (-5/8)^{5/3} = [(-5/8)^{1/3}]^5 = [ - (5/8)^{1/3} ]^5 = - (5/8)^{5/3} ≈ - (0.855)^5 ≈ -0.46, floor is -1. Therefore, same as before. Therefore, even with these x, floor is still -1 or 0. Therefore, the range remains { -1, 0, 1 }.

Therefore, regardless of whether x is integer or non-integer (as long as the expression is real), the floor function only takes values -1, 0, or 1. Therefore, the number of distinct values in the range is 3. Therefore, the answer is 3. Therefore, I think the answer is \boxed{3}.

But just to be thorough, let me check if there's any x where (-5/8)^x is exactly an integer. For example, suppose there exists some x where (-5/8)^x is integer. For x = 0, it's 1. For x = 1, it's -5/8, not integer. For x = 2, 25/64, not integer. For x = 3, -125/512, not integer. For x = 4, 625/4096, not integer. So, no other integer values. Therefore, only x=0 gives an integer. Therefore, floor is 1. For other x, floor is -1 or 0. Therefore, three distinct values. Therefore, answer is 3.

Therefore, I think the answer is 3. However, let me check once more for x=0. The problem says x is in [0, ∞). So x=0 is included. Therefore, f(0) = floor(1) = 1. Then for x>0, floor is -1 or 0. Therefore, three distinct values. Therefore, the answer is 3. Therefore, I think the answer is \boxed{3}.

**Final Answer**
\boxed{3}

To determine how many **distinct values** exist in the range of the function
$$
f(x) = \left\lfloor \left(-\frac{5}{8}\right)^x \right\rfloor
$$
for $ x \in [0, \infty) $, we must first understand the behavior of the expression inside the floor function:
$$
\left(-\frac{5}{8}\right)^x
$$

---

### Step 1: Understanding the Domain

The expression $ \left(-\frac{5}{8}\right)^x $ is **not real** for most real values of $ x $, because raising a negative number to a non-integer power results in a **complex number**. However, the problem states that $ f(x) $ is defined for all $ x \in [0, \infty) $ such that $ f(x) $ is a real number. This implies that the function is only defined for values of $ x $ for which $ \left(-\frac{5}{8}\right)^x $ is real.

This happens **only when $ x $ is an integer**. For non-integer $ x $, the expression becomes complex, and thus the floor function is not defined in the real number system. Therefore, the **domain of $ f(x) $ is the set of non-negative integers**.

---

### Step 2: Evaluate the Function at Integer Values

We now evaluate $ f(x) = \left\lfloor \left(-\frac{5}{8}\right)^x \right\rfloor $ for $ x = 0, 1, 2, 3, \dots $:

- $ x = 0 $:
  $$
  \left(-\frac{5}{8}\right)^0 = 1 \Rightarrow \left\lfloor 1 \right\rfloor = 1
  $$

- $ x = 1 $:
  $$
  \left(-\frac{5}{8}\right)^1 = -\frac{5}{8} = -0.625 \Rightarrow \left\lfloor -0.625 \right\rfloor = -1
  $$

- $ x = 2 $:
  $$
  \left(-\frac{5}{8}\right)^2 = \frac{25}{64} \approx 0.390625 \Rightarrow \left\lfloor 0.390625 \right\rfloor = 0
  $$

- $ x = 3 $:
  $$
  \left(-\frac{5}{8}\right)^3 = -\frac{125}{512} \approx -0.24414 \Rightarrow \left\lfloor -0.24414 \right\rfloor = -1
  $$

- $ x = 4 $:
  $$
  \left(-\frac{5}{8}\right)^4 = \frac{625}{4096} \approx 0.15258789 \Rightarrow \left\lfloor 0.15258789 \right\rfloor = 0
  $$

- $ x = 5 $:
  $$
  \left(-\frac{5}{8}\right)^5 = -\frac{3125}{32768} \approx -0.09536743 \Rightarrow \left\lfloor -0.09536743 \right\rfloor = -1
  $$

- $ x = 6 $:
  $$
  \left(-\frac{5}{8}\right)^6 = \frac{15625}{262144} \approx 0.05960464 \Rightarrow \left\lfloor 0.05960464 \right\rfloor = 0
  $$

We observe a clear pattern:

- For **even** $ x $, $ \left(-\frac{5}{8}\right)^x > 0 $ and less than 1 → floor is **0**.
- For **odd** $ x $, $ \left(-\frac{5}{8}\right)^x < 0 $ and greater than -1 → floor is **-1**.
- At $ x = 0 $, the value is exactly **1**, so floor is **1**.

---

### Step 3: Determine the Range

From the above, the function $ f(x) $ takes on the following values:

- $ f(0) = 1 $
- $ f(x) = -1 $ for all odd $ x \geq 1 $
- $ f(x) = 0 $ for all even $ x \geq 2 $

Thus, the **range** of $ f(x) $ is:
$$
\{ -1, 0, 1 \}
$$

These are **three distinct values**.

---

### Final Answer

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many distinct values the function $ f(x) = \left\lfloor \left(-\frac{5}{8}\right)^x \right\rfloor $ can take when $ x $ is in the interval $[0, \infty)$. Alright, let me start by recalling what the floor function does. The floor function $ \lfloor y \rfloor $ gives the greatest integer less than or equal to $ y $. So, for any real number $ y $, $ \lfloor y \rfloor $ is the integer part of $ y $, essentially rounding down to the nearest integer.

Now, the function inside the floor is $ \left(-\frac{5}{8}\right)^x $. Let me think about how this behaves as $ x $ varies from 0 to infinity. First, let me note that $ -\frac{5}{8} $ is a negative number with absolute value less than 1. So, raising it to different powers will have different behaviors depending on whether the exponent is an integer or not. But since $ x $ is in $[0, \infty)$, it can be any non-negative real number, not just integers. Hmm, so I need to consider real exponents here.

Wait, but when dealing with exponents, if the base is negative, we have to be careful. For example, $ (-a)^x $ is only real if $ x $ is an integer or if $ a $ is positive and $ x $ is rational with an odd denominator, etc. But since $ x $ is a real number here, unless $ x $ is an integer, $ (-\frac{5}{8})^x $ might not be real. Wait, but the problem says that $ f(x) $ is defined for all $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number. So maybe the function is only defined for those $ x $ where $ (-\frac{5}{8})^x $ is real? But the problem says it's defined for all $ x $ in $[0, \infty)$. Hmm, maybe there's a misunderstanding here.

Wait, actually, if $ x $ is a real number, then $ (-\frac{5}{8})^x $ is not real unless $ x $ is an integer. Because if $ x $ is not an integer, then taking a negative number to a non-integer power would result in a complex number. However, the problem states that $ f(x) $ is defined for all $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number. So maybe the function is only defined for integer values of $ x $? But the problem says "defined for all values of $ x $ in $[0, \infty)$", which is confusing. Wait, maybe there's a mistake here?

Wait, let me check again. The problem says: "Let $ f(x) = \left\lfloor \left(-\frac{5}{8}\right)^x \right\rfloor $ be a function that is defined for all values of $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number." So, perhaps the function is only defined for those $ x $ where $ \left(-\frac{5}{8}\right)^x $ is real. But for real numbers, $ (-\frac{5}{8})^x $ is real only when $ x $ is an integer or when $ x $ is a rational number with an odd denominator? Wait, but even so, for real exponents, unless the base is positive, it's not real. So, for example, $ (-\frac{5}{8})^x $ is real only when $ x $ is an integer? Because otherwise, if $ x $ is not an integer, you can't take a negative number to a non-integer power and get a real number. Therefore, maybe the function is only defined for integer values of $ x $?

But the problem says "defined for all values of $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number." So maybe the domain is actually all real numbers $ x \geq 0 $, but only when $ (-\frac{5}{8})^x $ is real. However, since $ (-\frac{5}{8})^x $ is real only when $ x $ is an integer? Wait, but even for non-integer exponents, if you write it as $ e^{x \ln(-5/8)} $, but the logarithm of a negative number is not real. Therefore, $ (-\frac{5}{8})^x $ is not real for non-integer $ x $. Therefore, the function $ f(x) $ is only defined for integer values of $ x $ in $[0, \infty)$. Therefore, the domain is actually the non-negative integers. Therefore, the problem is actually about the function $ f(x) = \lfloor (-5/8)^x \rfloor $ where $ x $ is a non-negative integer. Therefore, the question is asking for the number of distinct values in the range of this function when $ x $ is a non-negative integer.

But the problem says "defined for all values of $ x $ in $[0, \infty)$ such that $ f(x) $ is a real number." So maybe the problem is considering $ x $ as real numbers, but only when the expression is real. However, since for real $ x $, the expression is real only when $ x $ is an integer. Therefore, the domain is actually the non-negative integers. Therefore, the problem is equivalent to considering $ x $ as non-negative integers. Therefore, I need to evaluate $ f(x) = \lfloor (-5/8)^x \rfloor $ for $ x = 0, 1, 2, 3, ... $, and find how many distinct values there are.

Alternatively, maybe there's a different interpretation. Maybe the problem is considering complex numbers, but since the floor function is defined for real numbers, and the problem says $ f(x) $ is a real number, so the expression inside the floor must be real. Therefore, the domain is only those $ x $ for which $ (-5/8)^x $ is real. Which, as discussed, is only when $ x $ is an integer. Therefore, the domain is non-negative integers. Therefore, the problem reduces to considering integer exponents.

Therefore, proceeding with that assumption, let me consider $ x $ as non-negative integers. Let me compute $ (-5/8)^x $ for $ x = 0, 1, 2, 3, ... $, then take the floor of each, and see how many distinct values there are.

Starting with $ x = 0 $: $ (-5/8)^0 = 1 $. Therefore, $ f(0) = \lfloor 1 \rfloor = 1 $.

For $ x = 1 $: $ (-5/8)^1 = -5/8 = -0.625 $. The floor of -0.625 is -1, since -1 is the greatest integer less than or equal to -0.625. Therefore, $ f(1) = -1 $.

For $ x = 2 $: $ (-5/8)^2 = (25/64) ≈ 0.390625 $. The floor of that is 0. So $ f(2) = 0 $.

For $ x = 3 $: $ (-5/8)^3 = -125/512 ≈ -0.244140625 $. The floor of that is -1. So $ f(3) = -1 $.

For $ x = 4 $: $ (-5/8)^4 = 625/4096 ≈ 0.152587890625 $. Floor is 0. So $ f(4) = 0 $.

For $ x = 5 $: $ (-5/8)^5 = -3125/32768 ≈ -0.095367431640625 $. Floor is -1. So $ f(5) = -1 $.

For $ x = 6 $: $ (-5/8)^6 = 15625/262144 ≈ 0.059604644775390625 $. Floor is 0. So $ f(6) = 0 $.

Continuing this pattern, it seems that for even exponents, the value is positive and between 0 and 1, so floor is 0. For odd exponents, the value is negative and between -1 and 0, so floor is -1. However, let me check for higher exponents to see if this pattern continues.

For $ x = 7 $: $ (-5/8)^7 ≈ -0.07421875 $. Wait, no, let me compute it properly. Wait, $ (-5/8)^7 = - (5/8)^7. Let me compute $ (5/8)^7 $:

$ 5^7 = 78125 $, $ 8^7 = 2097152 $. So $ 78125 / 2097152 ≈ 0.03725290298... $. Therefore, $ (-5/8)^7 ≈ -0.0372529... $. So floor is -1. Similarly, for $ x = 8 $: $ (5/8)^8 ≈ 0.02685546875 $, so floor is 0. So yes, the pattern continues.

Therefore, for even $ x $, $ (-5/8)^x $ is positive and less than 1, so floor is 0. For odd $ x $, $ (-5/8)^x $ is negative and greater than -1, so floor is -1. However, wait, let me check for $ x = 0 $, which is even, and gives 1, which floor is 1. So, for $ x = 0 $, it's 1, and for all other even $ x \geq 2 $, it's between 0 and 1, so floor is 0. For odd $ x \geq 1 $, it's between -1 and 0, so floor is -1. Therefore, the function $ f(x) $ takes on three values: 1, 0, and -1. Wait, but let me check if there are any other values.

Wait, for $ x = 0 $, we have 1. For $ x = 1 $, -1. For $ x = 2 $, 0. For $ x = 3 $, -1. For $ x = 4 $, 0. Etc. So the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3? But let me check again.

Wait, but maybe there's a case where $ (-5/8)^x $ is exactly an integer? For example, when is $ (-5/8)^x $ an integer? For $ x = 0 $, it's 1, which is integer. For other $ x $, since $ 5/8 $ is a fraction, raising it to a positive integer power will give a fraction between 0 and 1 (for even exponents) or between -1 and 0 (for odd exponents). Therefore, the only integer value is 1 at $ x = 0 $. For all other $ x $, the value is between -1 and 1, but not including -1 or 1. Therefore, floor of those values would be 0 for positive values (since they are between 0 and 1) and -1 for negative values (since they are between -1 and 0). Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3?

But wait, let me check for $ x = 0 $, the value is 1, floor is 1. For $ x = 1 $, floor is -1. For $ x = 2 $, floor is 0. For $ x = 3 $, floor is -1. For $ x = 4 $, floor is 0. So yes, only three values. Therefore, the number of distinct values is 3. But let me check if there's any other possible value. For example, could $ (-5/8)^x $ ever be equal to 0? No, because $ (-5/8)^x $ approaches 0 as $ x $ approaches infinity, but never actually reaches 0. Similarly, it never reaches -1 or 1 except at $ x = 0 $ and $ x = 1 $? Wait, at $ x = 1 $, it's -5/8, which is -0.625, not -1. So floor is -1. So the only integer value is 1. Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3?

But wait, let me check for $ x = 0 $, which is allowed. So the function is defined for all $ x $ in $[0, \infty)$, but only when the expression is real. So if we take $ x $ as non-negative integers, then yes, the range is { -1, 0, 1 }, so three values. However, maybe there's a mistake here. Let me think again.

Wait, but maybe there's a case where $ (-5/8)^x $ is exactly an integer? For example, if $ x $ is such that $ (-5/8)^x $ is integer. Let me think. For $ x $ being a non-negative integer, $ (-5/8)^x $ is integer only when $ x = 0 $, because for $ x \geq 1 $, the denominator is 8^x, which is 2^{3x}, and the numerator is (-5)^x. So unless 8^x divides (-5)^x, which is only possible if x = 0. Therefore, only at x = 0, the value is integer. Therefore, floor of that is 1. For other x, the value is a fraction between -1 and 1 (excluding the endpoints), so floor is -1 or 0. Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3. But let me check again for x=0,1,2,3,...:

x=0: 1 → floor 1

x=1: -5/8 ≈ -0.625 → floor -1

x=2: 25/64 ≈ 0.390625 → floor 0

x=3: -125/512 ≈ -0.244 → floor -1

x=4: 625/4096 ≈ 0.1525 → floor 0

x=5: -3125/32768 ≈ -0.095 → floor -1

x=6: 15625/262144 ≈ 0.0596 → floor 0

x=7: -78125/2097152 ≈ -0.03725 → floor -1

x=8: 390625/16777216 ≈ 0.02328 → floor 0

So yes, the pattern continues. So the function takes values 1, -1, 0. Therefore, three distinct values. Therefore, the answer is 3. But wait, the problem says "the range of f(x)", so maybe there's something I'm missing here?

Wait, but let me check if there's any other possible value. For example, could the floor function ever give something else? For example, if $ (-5/8)^x $ is between 0 and 1, floor is 0. If it's between -1 and 0, floor is -1. If it's exactly 1, floor is 1. If it's exactly -1, floor is -1. But $ (-5/8)^x $ is never exactly -1 or 1 except at x=0. Because for x=0, it's 1. For x=1, it's -5/8, which is not -1. For x approaching infinity, $ (-5/8)^x $ approaches 0. So, the only integer value is 1. Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3. But let me check again.

Wait, but maybe there's a case where $ (-5/8)^x $ is equal to 0? But no, since $ (-5/8)^x $ approaches 0 as x approaches infinity, but never actually reaches 0. Therefore, floor of values approaching 0 from the positive side is 0, and from the negative side is -1. Therefore, the only integer value is 1. Therefore, the range is { -1, 0, 1 }, so three distinct values. Therefore, the answer is 3. But let me check again for x=0, which is included in the domain. So yes, x=0 is allowed, and gives 1. So the answer is 3. Therefore, the number of distinct values is 3. Therefore, the answer is \boxed{3}?

Wait, but let me check again. Maybe there's a mistake in assuming that the domain is only integers. The problem says "defined for all values of x in [0, ∞) such that f(x) is a real number." So, if x is not an integer, then (-5/8)^x is not real. Therefore, the domain is only the non-negative integers. Therefore, the function is only defined at integer points. Therefore, the range is { -1, 0, 1 }, so three values. Therefore, answer is 3. But maybe the problem is considering x as real numbers, but somehow (-5/8)^x is real? Wait, but how?

Wait, another thought: if x is a rational number with an odd denominator, then (-5/8)^x can be real. For example, if x = 1/3, then (-5/8)^{1/3} is the cube root of -5/8, which is real. Similarly, x = 2/3 would be the cube root of (25/64), which is real. However, if x is irrational, then (-5/8)^x is not real. Therefore, if the problem allows x to be any real number where (-5/8)^x is real, then the domain would include more points. However, the problem says "defined for all values of x in [0, ∞) such that f(x) is a real number." So, the domain is all x in [0, ∞) for which (-5/8)^x is real. Therefore, the domain is all x in [0, ∞) where x is rational with odd denominator in reduced form, or x is integer. But this complicates things. However, the problem says "the function is defined for all values of x in [0, ∞) such that f(x) is a real number." So, maybe the problem is considering x as real numbers, but only when the expression is real. However, in that case, the function is defined on a dense set of points (like all rationals with odd denominators), but the problem is asking for the range. However, if x can be any real number where (-5/8)^x is real, then the function's range might be different. However, this seems more complicated, and the problem might be intended for integer exponents. But since the problem didn't specify that x is an integer, but said "defined for all values of x in [0, ∞) such that f(x) is a real number," maybe we need to consider real exponents where the expression is real.

But how can we handle that? Let me think. For real exponents, (-5/8)^x is real if and only if x is a rational number with an odd denominator when written in lowest terms, or x is an integer. However, since the problem says "defined for all values of x in [0, ∞)", but only when the expression is real. Therefore, the domain is the set of real numbers x ≥ 0 for which (-5/8)^x is real. However, this set is not straightforward. For example, for x irrational, it's not real. For x rational with even denominator, it's not real. For x rational with odd denominator, it's real. For x integer, it's real. Therefore, the domain is a countable dense subset of [0, ∞). However, the problem is asking for the range of f(x). If we consider all such x, then the function's range might be different. However, this seems complicated. But maybe the problem is intended for integer exponents. Since otherwise, the problem becomes more complex and the answer might not be straightforward.

Alternatively, maybe the problem is considering complex numbers, but since the floor function is defined for real numbers, and the problem says f(x) is a real number, so the expression inside must be real. Therefore, the domain is as discussed. However, if we consider x as real numbers where (-5/8)^x is real, then for x in [0, ∞), the expression is real only when x is an integer or x is a rational number with odd denominator. However, even so, the function's range would depend on the values of x. However, this seems too complicated for a problem that is likely expecting an answer of 3. Therefore, maybe the problem is intended for integer exponents. Therefore, proceeding with that assumption, the answer is 3. However, let me check again.

Alternatively, maybe the problem is considering x as real numbers, and interpreting (-5/8)^x as e^{x ln(-5/8)}, but since ln(-5/8) is not real, this is not possible. Therefore, the only way for (-5/8)^x to be real is if x is an integer. Therefore, the domain is integers. Therefore, the answer is 3. Therefore, I think the answer is 3. However, let me check once more.

Wait, but if x is allowed to be any real number where (-5/8)^x is real, then for example, x = 1/3, which is allowed, then (-5/8)^{1/3} is real. Let me compute that. The cube root of -5/8 is - (5/8)^{1/3} ≈ -0.855. Therefore, floor of that is -1. Similarly, for x = 2/3, (-5/8)^{2/3} is the cube root of (25/64) ≈ 0.7937, floor is 0. For x = 1/2, which is not allowed because denominator is even, so (-5/8)^{1/2} is not real. For x = 3/2, which is not allowed because denominator is even. Wait, x = 3/2 is 1.5, which is not an integer, and denominator is 2, which is even. Therefore, (-5/8)^{3/2} is not real. Therefore, only for x with odd denominators in reduced form. For example, x = 1/3, 2/3, 1/5, 2/5, etc. So, for such x, the value of (-5/8)^x is real. Therefore, the domain is more than just integers. Therefore, the function is defined on more points. Therefore, the range might include more values. Therefore, maybe my previous conclusion of 3 is incorrect.

Therefore, I need to consider this possibility. Let me think. If x is allowed to be any real number where (-5/8)^x is real, then for x in [0, ∞), the expression is real when x is an integer or x is a rational number with odd denominator. However, even so, the function's range might have more values. For example, take x = 1/3, which gives (-5/8)^{1/3} ≈ -0.855, floor is -1. Similarly, x = 2/3 gives ≈ 0.7937, floor is 0. x = 1/5: (-5/8)^{1/5} is the fifth root of -5/8 ≈ -0.86, floor is -1. x = 2/5: fifth root of (25/64) ≈ 0.83, floor is 0. Similarly, for x = 3/5: fifth root of (-5/8)^3 = fifth root of -125/512 ≈ -0.62, floor is -1. x = 4/5: fifth root of (5/8)^4 ≈ 0.59, floor is 0. So, similar to integer exponents, for non-integer rational exponents with odd denominators, the value is between -1 and 0 or 0 and 1, so floor is -1 or 0. Except for x=0, which gives 1. Therefore, even if we consider these rational exponents, the floor function still only gives -1, 0, or 1. Therefore, the range is still { -1, 0, 1 }, so three values. Therefore, even if we consider more points in the domain, the range doesn't increase. Therefore, the answer is still 3.

But wait, let me check for some other x. For example, take x approaching 0 from the right. Let me consider x approaching 0. For example, x = 0.1. But x = 0.1 is 1/10, which has denominator 10, even. Therefore, (-5/8)^{0.1} is not real. Therefore, not allowed. How about x = 1/1, which is integer. x = 1/3, which is allowed. What about x = 1/1, which is 1. So, for x approaching 0 from the right, but only for x where it's allowed. For example, x approaching 0 from the right through rational numbers with odd denominators. For example, x = 1/n where n is odd. Let me take n approaching infinity, so x approaching 0. Then (-5/8)^{1/n} approaches 1, since any number to the power 0 is 1. Therefore, for x approaching 0, the value approaches 1. Therefore, floor of that approaches 1. However, for x = 0, it's exactly 1. For x approaching 0 from the right, the value is approaching 1 from below or above? Let me check for x = 1/n, n odd. For example, n = 1: x = 1, value is -5/8 ≈ -0.625. For n = 3: x = 1/3, value ≈ -0.855. For n = 5: x = 1/5, value ≈ -0.86. Wait, as n increases, x approaches 0, and (-5/8)^{1/n} approaches 1? Wait, no. Wait, (-5/8)^{1/n} is equal to e^{(ln(-5/8))/n}. But ln(-5/8) is not real. Wait, but if we take x = 1/n with n odd, then (-5/8)^{1/n} is real. Let me compute it as (-5/8)^{1/n} = - (5/8)^{1/n} because n is odd. Therefore, for x = 1/n, the value is - (5/8)^{1/n}. As n approaches infinity, (5/8)^{1/n} approaches 1, so the value approaches -1. Therefore, for x approaching 0 from the right through odd denominators, the value approaches -1. Therefore, floor of that approaches -1. However, at x = 0, the value is 1. Therefore, there is a jump from 1 at x=0 to approaching -1 as x approaches 0 from the right. Therefore, the floor function at x=0 is 1, and for x approaching 0 from the right, floor is -1. Therefore, the value 1 is only achieved at x=0. For other x, the floor is either -1 or 0. Therefore, the range is still { -1, 0, 1 }, so three values. Therefore, even considering non-integer x where the expression is real, the range doesn't increase. Therefore, the answer is 3.

But let me check for some other x. For example, take x = 2/3. Then (-5/8)^{2/3} is equal to [(-5/8)^{1/3}]^2 = [ - (5/8)^{1/3} ]^2 = (5/8)^{2/3} ≈ (0.855)^2 ≈ 0.731, floor is 0. Similarly, for x = 3/3 = 1, floor is -1. For x = 4/3, which is 1.333..., which is allowed? Wait, x = 4/3. Let me check if it's allowed. 4/3 is a rational number with denominator 3, which is odd. Therefore, yes, allowed. Then (-5/8)^{4/3} = [(-5/8)^{1/3}]^4 = [ - (5/8)^{1/3} ]^4 = (5/8)^{4/3} ≈ (0.855)^4 ≈ 0.53, floor is 0. Similarly, for x = 5/3, which is 1.666..., allowed. (-5/8)^{5/3} = [(-5/8)^{1/3}]^5 = [ - (5/8)^{1/3} ]^5 = - (5/8)^{5/3} ≈ - (0.855)^5 ≈ -0.46, floor is -1. Therefore, same as before. Therefore, even with these x, floor is still -1 or 0. Therefore, the range remains { -1, 0, 1 }.

Therefore, regardless of whether x is integer or non-integer (as long as the expression is real), the floor function only takes values -1, 0, or 1. Therefore, the number of distinct values in the range is 3. Therefore, the answer is 3. Therefore, I think the answer is \boxed{3}.

But just to be thorough, let me check if there's any x where (-5/8)^x is exactly an integer. For example, suppose there exists some x where (-5/8)^x is integer. For x = 0, it's 1. For x = 1, it's -5/8, not integer. For x = 2, 25/64, not integer. For x = 3, -125/512, not integer. For x = 4, 625/4096, not integer. So, no other integer values. Therefore, only x=0 gives an integer. Therefore, floor is 1. For other x, floor is -1 or 0. Therefore, three distinct values. Therefore, answer is 3.

Therefore, I think the answer is 3. However, let me check once more for x=0. The problem says x is in [0, ∞). So x=0 is included. Therefore, f(0) = floor(1) = 1. Then for x>0, floor is -1 or 0. Therefore, three distinct values. Therefore, the answer is 3. Therefore, I think the answer is \boxed{3}.

**Final Answer**
\boxed{3}

To determine how many **distinct values** exist in the range of the function
$$
f(x) = \left\lfloor \left(-\frac{5}{8}\right)^x \right\rfloor
$$
for $ x \in [0, \infty) $, we must first understand the behavior of the expression inside the floor function:
$$
\left(-\frac{5}{8}\right)^x
$$

---

### Step 1: Understanding the Domain

The expression $ \left(-\frac{5}{8}\right)^x $ is **not real** for most real values of $ x $, because raising a negative number to a non-integer power results in a **complex number**. However, the problem states that $ f(x) $ is defined for all $ x \in [0, \infty) $ such that $ f(x) $ is a real number. This implies that the function is only defined for values of $ x $ for which $ \left(-\frac{5}{8}\right)^x $ is real.

This happens **only when $ x $ is an integer**. For non-integer $ x $, the expression becomes complex, and thus the floor function is not defined in the real number system. Therefore, the **domain of $ f(x) $ is the set of non-negative integers**.

---

### Step 2: Evaluate the Function at Integer Values

We now evaluate $ f(x) = \left\lfloor \left(-\frac{5}{8}\right)^x \right\rfloor $ for $ x = 0, 1, 2, 3, \dots $:

- $ x = 0 $:
  $$
  \left(-\frac{5}{8}\right)^0 = 1 \Rightarrow \left\lfloor 1 \right\rfloor = 1
  $$

- $ x = 1 $:
  $$
  \left(-\frac{5}{8}\right)^1 = -\frac{5}{8} = -0.625 \Rightarrow \left\lfloor -0.625 \right\rfloor = -1
  $$

- $ x = 2 $:
  $$
  \left(-\frac{5}{8}\right)^2 = \frac{25}{64} \approx 0.390625 \Rightarrow \left\lfloor 0.390625 \right\rfloor = 0
  $$

- $ x = 3 $:
  $$
  \left(-\frac{5}{8}\right)^3 = -\frac{125}{512} \approx -0.24414 \Rightarrow \left\lfloor -0.24414 \right\rfloor = -1
  $$

- $ x = 4 $:
  $$
  \left(-\frac{5}{8}\right)^4 = \frac{625}{4096} \approx 0.15258789 \Rightarrow \left\lfloor 0.15258789 \right\rfloor = 0
  $$

- $ x = 5 $:
  $$
  \left(-\frac{5}{8}\right)^5 = -\frac{3125}{32768} \approx -0.09536743 \Rightarrow \left\lfloor -0.09536743 \right\rfloor = -1
  $$

- $ x = 6 $:
  $$
  \left(-\frac{5}{8}\right)^6 = \frac{15625}{262144} \approx 0.05960464 \Rightarrow \left\lfloor 0.05960464 \right\rfloor = 0
  $$

We observe a clear pattern:

- For **even** $ x $, $ \left(-\frac{5}{8}\right)^x > 0 $ and less than 1 → floor is **0**.
- For **odd** $ x $, $ \left(-\frac{5}{8}\right)^x < 0 $ and greater than -1 → floor is **-1**.
- At $ x = 0 $, the value is exactly **1**, so floor is **1**.

---

### Step 3: Determine the Range

From the above, the function $ f(x) $ takes on the following values:

- $ f(0) = 1 $
- $ f(x) = -1 $ for all odd $ x \geq 1 $
- $ f(x) = 0 $ for all even $ x \geq 2 $

Thus, the **range** of $ f(x) $ is:
$$
\{ -1, 0, 1 \}
$$

These are **three distinct values**.

---

### Final Answer

$$
\boxed{3}
$$

Solution: Since $-\frac58$ is a negative number, $f(x)$ is only defined for integer values of $x$, and will alternate between positive and negative values. Additionally, $\left|-\frac58\right|< 1$, so $|f(x)|$ will continually decrease and approach 0 as $x$ increases in the interval $x\ge0$. Therefore, the largest positive value will occur at $x=0$, giving us the positive upper bound of $\left\lfloor\left(-\frac58\right)^0\right\rfloor=1$. The negative value that is greatest in magnitude then occurs at the next integer value of $x$: $x=1$, giving us the negative lower bound of $\left\lfloor\left(-\frac58\right)^1\right\rfloor=-1$. This tells us that $-1 \le f(x) \le 1$. Since the $f(x)$ must be an integer, the only possible distinct values contained in the range are -1, 0, and 1. This gives us a total of $\boxed{3}$ values of $f(x)$ when $x\ge0$.


Explanation:
Okay, so I need to figure out how many ways we can form a subcommittee of 5 members from a Senate committee that has 8 Republicans and 6 Democrats. The catch is that the subcommittee must have at least one member from each party. Hmm, let me think about how to approach this.

First, I remember that when dealing with combinations and restrictions like "at least one from each party," sometimes it's easier to calculate the total number of possible subcommittees without any restrictions and then subtract the ones that don't meet the criteria. That is, subtract the subcommittees that are all Republicans or all Democrats. That should leave me with the number of subcommittees that have at least one from each party. Let me check if that makes sense.

So, the total number of ways to choose 5 members from the entire committee, regardless of party, would be the combination of 14 people taken 5 at a time. Wait, the committee has 8 Republicans and 6 Democrats, so total members are 8 + 6 = 14. Yeah, so total ways is C(14, 5). Then, subtract the number of all-Republican subcommittees and all-Democrat subcommittees.

Let me write that down:

Total subcommittees = C(14, 5)

Invalid subcommittees (all Republicans or all Democrats) = C(8, 5) + C(6, 5)

Therefore, the answer should be Total - Invalid = C(14,5) - C(8,5) - C(6,5).

But let me make sure I didn't miss anything. The problem says "at least one member from each party," so we need to exclude any subcommittee that is entirely one party. That seems right.

Alternatively, I could approach it by considering the different possible splits of Republicans and Democrats in the subcommittee. Since we need at least one from each party, the possible splits are:

- 1 Republican and 4 Democrats

- 2 Republicans and 3 Democrats

- 3 Republicans and 2 Democrats

- 4 Republicans and 1 Democrat

These are the only possible splits because the subcommittee has 5 members, and we can't have 0 Republicans or 0 Democrats. So, maybe I can calculate each of these cases separately and add them up. Let me try that method as well to verify if I get the same answer.

So, for each case:

Case 1: 1R and 4D. The number of ways is C(8,1) * C(6,4)

Case 2: 2R and 3D. C(8,2) * C(6,3)

Case 3: 3R and 2D. C(8,3) * C(6,2)

Case 4: 4R and 1D. C(8,4) * C(6,1)

Then, adding all these up should give the total number of valid subcommittees.

Let me compute both methods and see if they match. If they do, that gives me confidence that my answer is correct.

First, let me compute the first method:

Total subcommittees: C(14,5). Let me calculate that.

C(14,5) = 14! / (5! * 9!) = (14*13*12*11*10)/(5*4*3*2*1) = let me compute that step by step.

14*13 = 182; 182*12 = 2184; 2184*11 = 24024; 24024*10 = 240240. Then divide by 120 (which is 5!).

240240 / 120. Let me divide 240240 by 10 first: 24024. Then divide by 12: 24024 / 12 = 2002. Wait, 12*2000 = 24000, so 24024 - 24000 = 24, so 24/12 = 2. So total is 2000 + 2 = 2002. So C(14,5) is 2002.

Now, the invalid subcommittees:

All Republicans: C(8,5). Let me compute that. C(8,5) = C(8,3) because C(n,k) = C(n, n-k). So C(8,3) = 8*7*6 / 3*2*1 = 56.

Wait, 8*7 is 56, 56*6 is 336, divided by 6 (since 3*2*1=6). 336 / 6 = 56. Yes, that's correct.

All Democrats: C(6,5) = C(6,1) = 6.

Therefore, invalid subcommittees total 56 + 6 = 62.

Therefore, the answer via first method is 2002 - 62 = 1940.

Now let me check with the second method, calculating each case.

Case 1: 1R and 4D. C(8,1)*C(6,4).

C(8,1) is 8. C(6,4) is C(6,2) = 15. So 8*15 = 120.

Case 2: 2R and 3D. C(8,2)*C(6,3).

C(8,2) is 28. C(6,3) is 20. So 28*20 = 560.

Case 3: 3R and 2D. C(8,3)*C(6,2).

C(8,3) is 56. C(6,2) is 15. So 56*15 = 840.

Case 4: 4R and 1D. C(8,4)*C(6,1).

C(8,4) is 70. C(6,1) is 6. So 70*6 = 420.

Now, adding all these up: 120 + 560 + 840 + 420. Let me compute step by step.

120 + 560 = 680.

680 + 840 = 1520.

1520 + 420 = 1940.

Same result as the first method! So that's reassuring. Therefore, the answer is 1940.

But just to make sure I didn't make any calculation errors, let me verify each of the combination numbers again.

First, C(14,5) = 2002. Let me check with another approach. For example, I know that C(14,5) is 2002. Let me check with a calculator formula. Alternatively, I can recall that C(14,5) is indeed 2002.

For C(8,5) = 56. Yes, because 8 choose 5 is 56. And C(6,5) is 6. So 56 + 6 = 62. 2002 - 62 = 1940.

For the second method:

Case 1: 1R and 4D. C(8,1) is 8. C(6,4) is 15. 8*15=120. Correct.

Case 2: 2R and 3D. C(8,2) is 28. C(6,3) is 20. 28*20=560. Correct.

Case 3: 3R and 2D. C(8,3)=56, C(6,2)=15. 56*15=840. Correct.

Case 4: 4R and 1D. C(8,4)=70, C(6,1)=6. 70*6=420. Correct.

Adding them up: 120 + 560 is 680, plus 840 is 1520, plus 420 is 1940. Yep, same as before.

Therefore, I think 1940 is the correct answer.

But just to be thorough, let me check if there's another way to think about this problem. For example, using inclusion-exclusion. Wait, that's essentially what the first method was.

Alternatively, maybe I can think of generating functions? Although that might be overcomplicating, but let me try.

The generating function for Republicans is (1 + x)^8, and for Democrats is (1 + x)^6. The coefficient of x^5 in the product (1 + x)^8 * (1 + x)^6 = (1 + x)^14 would be C(14,5), which is the total number of subcommittees. But if we want to exclude the cases with all Republicans or all Democrats, we can subtract those. Which again is the same as the first method. So generating functions also lead to the same result.

Alternatively, if I wanted to use generating functions to directly count the valid subcommittees, I would need to consider the generating function where we have at least one Republican and at least one Democrat. So, for Republicans, it would be (x + x^2 + ... + x^8) and for Democrats (x + x^2 + ... + x^6). Then multiply them and find the coefficient of x^5.

But that might be more complicated, but let me try.

The generating function for Republicans with at least 1 and at most 8 is x*(1 + x + ... + x^7) = x*( (1 - x^8)/(1 - x) )

Similarly for Democrats: x*(1 + x + ... + x^5) = x*( (1 - x^6)/(1 - x) )

Multiplying these together: x^2 * (1 - x^8)(1 - x^6)/(1 - x)^2

We need the coefficient of x^5 in this expression. Which would be the same as the coefficient of x^(5 - 2) = x^3 in (1 - x^8)(1 - x^6)/(1 - x)^2.

But expanding (1 - x^8)(1 - x^6) = 1 - x^6 - x^8 + x^{14}

Therefore, the coefficient of x^3 in [1 - x^6 - x^8 + x^{14}] / (1 - x)^2.

Since we are looking for x^3, the terms -x^6, -x^8, and +x^{14} will not contribute because their exponents are higher than 3. Therefore, it's just the coefficient of x^3 in 1/(1 - x)^2, which is C(3 + 2 - 1, 2 - 1) = C(4,1) = 4. Wait, but that seems way too low. Wait, no, hold on. Wait, the coefficient of x^k in 1/(1 - x)^2 is C(k + 1, 1) = k + 1. So for k = 3, it's 4. But that would mean the total number is 4? But that contradicts our previous answer of 1940. So clearly, I made a mistake here.

Wait, why? Because I think I messed up the generating function approach. Let me check again.

Wait, the generating function for the number of subcommittees with at least one Republican and at least one Democrat is:

For Republicans: since we need at least 1, it's (number of ways to choose Republicans) which is sum_{r=1}^8 C(8, r) x^r. Similarly for Democrats: sum_{d=1}^6 C(6, d) x^d.

Therefore, the generating function is [sum_{r=1}^8 C(8, r) x^r] * [sum_{d=1}^6 C(6, d) x^d].

But the coefficient of x^5 in this product would be the number of ways to choose r Republicans and d Democrats such that r + d = 5, with r ≥ 1, d ≥ 1. Which is exactly the same as our second method. So, in other words, the coefficient of x^5 is the sum over r=1 to 4 of C(8, r)*C(6, 5 - r). Which is exactly the same as the cases we considered earlier.

But when I tried to use generating functions with the generating function expressions, I think I made an error in the algebra. Let me try again.

The generating function for Republicans with at least 1 is x*(1 + x + ... + x^7) = x*( (1 - x^8)/(1 - x) )

Similarly for Democrats: x*( (1 - x^6)/(1 - x) )

Multiplying them gives x^2*(1 - x^8 - x^6 + x^{14}) / (1 - x)^2.

So, to find the coefficient of x^5 in this, we need the coefficient of x^{5 - 2} = x^3 in (1 - x^6 - x^8 + x^{14}) / (1 - x)^2.

But as mentioned, the terms -x^6, -x^8, and +x^{14} will not contribute to the coefficient of x^3. Therefore, it's the same as the coefficient of x^3 in 1/(1 - x)^2, which is C(3 + 2 - 1, 2 - 1) = C(4, 1) = 4? Wait, that can't be. Wait, no, the coefficient of x^k in 1/(1 - x)^n is C(k + n - 1, n - 1). So here, n = 2, so coefficient of x^3 is C(3 + 2 - 1, 2 - 1) = C(4, 1) = 4. But that would mean the total number of subcommittees is 4? Which is not possible.

Wait, clearly, there's a mistake here. Because when I expand the generating function, the coefficient of x^5 should be 1940, but according to this method, it's 4? That can't be. So where is the error?

Ah! Wait, no. Wait, the generating function [sum_{r=1}^8 C(8, r) x^r] * [sum_{d=1}^6 C(6, d) x^d] is equal to [ (1 + x)^8 - 1 ] * [ (1 + x)^6 - 1 ].

Therefore, expanding that, we have (1 + x)^8*(1 + x)^6 - (1 + x)^8 - (1 + x)^6 + 1.

Therefore, the coefficient of x^5 in this expression is C(14,5) - C(8,5) - C(6,5) + 1. Wait, but the +1 is the constant term, which doesn't affect the coefficient of x^5. Therefore, the coefficient of x^5 is C(14,5) - C(8,5) - C(6,5). Which is exactly the same as our first method. Therefore, the answer is 2002 - 56 - 6 = 1940.

So my earlier mistake was in the generating function approach when I tried to write it as x^2*(...). I think I confused the generating functions. So the correct way is to use the generating function as [ (1 + x)^8 - 1 ] * [ (1 + x)^6 - 1 ] and then find the coefficient of x^5, which gives the same result as the first method.

Therefore, all methods converge to 1940.

Just to make sure, let me check with actual numbers. Let me compute the total number of subcommittees with at least one from each party.

Alternatively, think of it as the total number of subcommittees minus those that are all Republicans or all Democrats.

Total subcommittees: 2002.

All Republicans: C(8,5) = 56.

All Democrats: C(6,5) = 6.

So 2002 - 56 - 6 = 1940.

Yes.

Alternatively, if I think of the possible splits:

1R and 4D: 8 * 15 = 120

2R and 3D: 28 * 20 = 560

3R and 2D: 56 * 15 = 840

4R and 1D: 70 * 6 = 420

Total: 120 + 560 + 840 + 420 = 1940.

Same answer.

Therefore, I think I can be confident that 1940 is the correct answer.

But just to make sure, let me check if there's any possibility that I miscalculated any of the combination numbers.

For example, C(8,1) is 8, correct. C(6,4) is 15. 8*15=120. Correct.

C(8,2) is 28. C(6,3) is 20. 28*20=560. Correct.

C(8,3)=56. C(6,2)=15. 56*15=840. Correct.

C(8,4)=70. C(6,1)=6. 70*6=420. Correct.

Adding them up: 120 + 560 is 680. 680 + 840 is 1520. 1520 + 420 is 1940. Correct.

Alternatively, if I think of C(8,5) = 56. Let me compute that again.

C(8,5) = 8! / (5! * 3!) = (8*7*6)/6 = 56. Correct.

C(6,5)=6. Correct.

Total subcommittees C(14,5)=2002. Correct.

So, all numbers check out.

Therefore, I think the answer is 1940.

**Final Answer**
The number of ways is \boxed{1940}.

To determine the number of ways to form a subcommittee of 5 members from a Senate committee consisting of 8 Republicans and 6 Democrats, with the condition that the subcommittee must include **at least one member from each party**, we can approach the problem using **combinatorics** and **inclusion-exclusion**.

---

### Step 1: Total Number of Subcommittees (Without Restrictions)

We are choosing 5 members from a total of 14 (8 Republicans + 6 Democrats). The total number of ways to choose 5 members from 14 is:

$$
\binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002
$$

---

### Step 2: Subtract Invalid Subcommittees

We now subtract the number of subcommittees that **do not meet the requirement** of having at least one member from each party. These are the subcommittees that are **all Republicans** or **all Democrats**.

- **All Republicans**: Choose 5 Republicans from 8:
  $$
  \binom{8}{5} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
  $$

- **All Democrats**: Choose 5 Democrats from 6:
  $$
  \binom{6}{5} = \binom{6}{1} = 6
  $$

So, the number of **invalid** subcommittees is:
$$
56 + 6 = 62
$$

---

### Step 3: Valid Subcommittees (At Least One from Each Party)

Now, subtract the invalid subcommittees from the total:
$$
2002 - 62 = 1940
$$

---

### Step 4: Verification via Case Analysis

We can also verify this by considering all valid combinations of Republicans and Democrats that sum to 5 members, with at least one from each party. The valid splits are:

- 1R + 4D: $ \binom{8}{1} \times \binom{6}{4} = 8 \times 15 = 120 $
- 2R + 3D: $ \binom{8}{2} \times \binom{6}{3} = 28 \times 20 = 560 $
- 3R + 2D: $ \binom{8}{3} \times \binom{6}{2} = 56 \times 15 = 840 $
- 4R + 1D: $ \binom{8}{4} \times \binom{6}{1} = 70 \times 6 = 420 $

Adding these:
$$
120 + 560 + 840 + 420 = 1940
$$

This confirms the result.

---

### Final Answer

$$
\boxed{1940}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many ways we can form a subcommittee of 5 members from a Senate committee that has 8 Republicans and 6 Democrats. The catch is that the subcommittee must have at least one member from each party. Hmm, let me think about how to approach this.

First, I remember that when dealing with combinations and restrictions like "at least one from each party," sometimes it's easier to calculate the total number of possible subcommittees without any restrictions and then subtract the ones that don't meet the criteria. That is, subtract the subcommittees that are all Republicans or all Democrats. That should leave me with the number of subcommittees that have at least one from each party. Let me check if that makes sense.

So, the total number of ways to choose 5 members from the entire committee, regardless of party, would be the combination of 14 people taken 5 at a time. Wait, the committee has 8 Republicans and 6 Democrats, so total members are 8 + 6 = 14. Yeah, so total ways is C(14, 5). Then, subtract the number of all-Republican subcommittees and all-Democrat subcommittees.

Let me write that down:

Total subcommittees = C(14, 5)

Invalid subcommittees (all Republicans or all Democrats) = C(8, 5) + C(6, 5)

Therefore, the answer should be Total - Invalid = C(14,5) - C(8,5) - C(6,5).

But let me make sure I didn't miss anything. The problem says "at least one member from each party," so we need to exclude any subcommittee that is entirely one party. That seems right.

Alternatively, I could approach it by considering the different possible splits of Republicans and Democrats in the subcommittee. Since we need at least one from each party, the possible splits are:

- 1 Republican and 4 Democrats

- 2 Republicans and 3 Democrats

- 3 Republicans and 2 Democrats

- 4 Republicans and 1 Democrat

These are the only possible splits because the subcommittee has 5 members, and we can't have 0 Republicans or 0 Democrats. So, maybe I can calculate each of these cases separately and add them up. Let me try that method as well to verify if I get the same answer.

So, for each case:

Case 1: 1R and 4D. The number of ways is C(8,1) * C(6,4)

Case 2: 2R and 3D. C(8,2) * C(6,3)

Case 3: 3R and 2D. C(8,3) * C(6,2)

Case 4: 4R and 1D. C(8,4) * C(6,1)

Then, adding all these up should give the total number of valid subcommittees.

Let me compute both methods and see if they match. If they do, that gives me confidence that my answer is correct.

First, let me compute the first method:

Total subcommittees: C(14,5). Let me calculate that.

C(14,5) = 14! / (5! * 9!) = (14*13*12*11*10)/(5*4*3*2*1) = let me compute that step by step.

14*13 = 182; 182*12 = 2184; 2184*11 = 24024; 24024*10 = 240240. Then divide by 120 (which is 5!).

240240 / 120. Let me divide 240240 by 10 first: 24024. Then divide by 12: 24024 / 12 = 2002. Wait, 12*2000 = 24000, so 24024 - 24000 = 24, so 24/12 = 2. So total is 2000 + 2 = 2002. So C(14,5) is 2002.

Now, the invalid subcommittees:

All Republicans: C(8,5). Let me compute that. C(8,5) = C(8,3) because C(n,k) = C(n, n-k). So C(8,3) = 8*7*6 / 3*2*1 = 56.

Wait, 8*7 is 56, 56*6 is 336, divided by 6 (since 3*2*1=6). 336 / 6 = 56. Yes, that's correct.

All Democrats: C(6,5) = C(6,1) = 6.

Therefore, invalid subcommittees total 56 + 6 = 62.

Therefore, the answer via first method is 2002 - 62 = 1940.

Now let me check with the second method, calculating each case.

Case 1: 1R and 4D. C(8,1)*C(6,4).

C(8,1) is 8. C(6,4) is C(6,2) = 15. So 8*15 = 120.

Case 2: 2R and 3D. C(8,2)*C(6,3).

C(8,2) is 28. C(6,3) is 20. So 28*20 = 560.

Case 3: 3R and 2D. C(8,3)*C(6,2).

C(8,3) is 56. C(6,2) is 15. So 56*15 = 840.

Case 4: 4R and 1D. C(8,4)*C(6,1).

C(8,4) is 70. C(6,1) is 6. So 70*6 = 420.

Now, adding all these up: 120 + 560 + 840 + 420. Let me compute step by step.

120 + 560 = 680.

680 + 840 = 1520.

1520 + 420 = 1940.

Same result as the first method! So that's reassuring. Therefore, the answer is 1940.

But just to make sure I didn't make any calculation errors, let me verify each of the combination numbers again.

First, C(14,5) = 2002. Let me check with another approach. For example, I know that C(14,5) is 2002. Let me check with a calculator formula. Alternatively, I can recall that C(14,5) is indeed 2002.

For C(8,5) = 56. Yes, because 8 choose 5 is 56. And C(6,5) is 6. So 56 + 6 = 62. 2002 - 62 = 1940.

For the second method:

Case 1: 1R and 4D. C(8,1) is 8. C(6,4) is 15. 8*15=120. Correct.

Case 2: 2R and 3D. C(8,2) is 28. C(6,3) is 20. 28*20=560. Correct.

Case 3: 3R and 2D. C(8,3)=56, C(6,2)=15. 56*15=840. Correct.

Case 4: 4R and 1D. C(8,4)=70, C(6,1)=6. 70*6=420. Correct.

Adding them up: 120 + 560 is 680, plus 840 is 1520, plus 420 is 1940. Yep, same as before.

Therefore, I think 1940 is the correct answer.

But just to be thorough, let me check if there's another way to think about this problem. For example, using inclusion-exclusion. Wait, that's essentially what the first method was.

Alternatively, maybe I can think of generating functions? Although that might be overcomplicating, but let me try.

The generating function for Republicans is (1 + x)^8, and for Democrats is (1 + x)^6. The coefficient of x^5 in the product (1 + x)^8 * (1 + x)^6 = (1 + x)^14 would be C(14,5), which is the total number of subcommittees. But if we want to exclude the cases with all Republicans or all Democrats, we can subtract those. Which again is the same as the first method. So generating functions also lead to the same result.

Alternatively, if I wanted to use generating functions to directly count the valid subcommittees, I would need to consider the generating function where we have at least one Republican and at least one Democrat. So, for Republicans, it would be (x + x^2 + ... + x^8) and for Democrats (x + x^2 + ... + x^6). Then multiply them and find the coefficient of x^5.

But that might be more complicated, but let me try.

The generating function for Republicans with at least 1 and at most 8 is x*(1 + x + ... + x^7) = x*( (1 - x^8)/(1 - x) )

Similarly for Democrats: x*(1 + x + ... + x^5) = x*( (1 - x^6)/(1 - x) )

Multiplying these together: x^2 * (1 - x^8)(1 - x^6)/(1 - x)^2

We need the coefficient of x^5 in this expression. Which would be the same as the coefficient of x^(5 - 2) = x^3 in (1 - x^8)(1 - x^6)/(1 - x)^2.

But expanding (1 - x^8)(1 - x^6) = 1 - x^6 - x^8 + x^{14}

Therefore, the coefficient of x^3 in [1 - x^6 - x^8 + x^{14}] / (1 - x)^2.

Since we are looking for x^3, the terms -x^6, -x^8, and +x^{14} will not contribute because their exponents are higher than 3. Therefore, it's just the coefficient of x^3 in 1/(1 - x)^2, which is C(3 + 2 - 1, 2 - 1) = C(4,1) = 4. Wait, but that seems way too low. Wait, no, hold on. Wait, the coefficient of x^k in 1/(1 - x)^2 is C(k + 1, 1) = k + 1. So for k = 3, it's 4. But that would mean the total number is 4? But that contradicts our previous answer of 1940. So clearly, I made a mistake here.

Wait, why? Because I think I messed up the generating function approach. Let me check again.

Wait, the generating function for the number of subcommittees with at least one Republican and at least one Democrat is:

For Republicans: since we need at least 1, it's (number of ways to choose Republicans) which is sum_{r=1}^8 C(8, r) x^r. Similarly for Democrats: sum_{d=1}^6 C(6, d) x^d.

Therefore, the generating function is [sum_{r=1}^8 C(8, r) x^r] * [sum_{d=1}^6 C(6, d) x^d].

But the coefficient of x^5 in this product would be the number of ways to choose r Republicans and d Democrats such that r + d = 5, with r ≥ 1, d ≥ 1. Which is exactly the same as our second method. So, in other words, the coefficient of x^5 is the sum over r=1 to 4 of C(8, r)*C(6, 5 - r). Which is exactly the same as the cases we considered earlier.

But when I tried to use generating functions with the generating function expressions, I think I made an error in the algebra. Let me try again.

The generating function for Republicans with at least 1 is x*(1 + x + ... + x^7) = x*( (1 - x^8)/(1 - x) )

Similarly for Democrats: x*( (1 - x^6)/(1 - x) )

Multiplying them gives x^2*(1 - x^8 - x^6 + x^{14}) / (1 - x)^2.

So, to find the coefficient of x^5 in this, we need the coefficient of x^{5 - 2} = x^3 in (1 - x^6 - x^8 + x^{14}) / (1 - x)^2.

But as mentioned, the terms -x^6, -x^8, and +x^{14} will not contribute to the coefficient of x^3. Therefore, it's the same as the coefficient of x^3 in 1/(1 - x)^2, which is C(3 + 2 - 1, 2 - 1) = C(4, 1) = 4? Wait, that can't be. Wait, no, the coefficient of x^k in 1/(1 - x)^n is C(k + n - 1, n - 1). So here, n = 2, so coefficient of x^3 is C(3 + 2 - 1, 2 - 1) = C(4, 1) = 4. But that would mean the total number of subcommittees is 4? Which is not possible.

Wait, clearly, there's a mistake here. Because when I expand the generating function, the coefficient of x^5 should be 1940, but according to this method, it's 4? That can't be. So where is the error?

Ah! Wait, no. Wait, the generating function [sum_{r=1}^8 C(8, r) x^r] * [sum_{d=1}^6 C(6, d) x^d] is equal to [ (1 + x)^8 - 1 ] * [ (1 + x)^6 - 1 ].

Therefore, expanding that, we have (1 + x)^8*(1 + x)^6 - (1 + x)^8 - (1 + x)^6 + 1.

Therefore, the coefficient of x^5 in this expression is C(14,5) - C(8,5) - C(6,5) + 1. Wait, but the +1 is the constant term, which doesn't affect the coefficient of x^5. Therefore, the coefficient of x^5 is C(14,5) - C(8,5) - C(6,5). Which is exactly the same as our first method. Therefore, the answer is 2002 - 56 - 6 = 1940.

So my earlier mistake was in the generating function approach when I tried to write it as x^2*(...). I think I confused the generating functions. So the correct way is to use the generating function as [ (1 + x)^8 - 1 ] * [ (1 + x)^6 - 1 ] and then find the coefficient of x^5, which gives the same result as the first method.

Therefore, all methods converge to 1940.

Just to make sure, let me check with actual numbers. Let me compute the total number of subcommittees with at least one from each party.

Alternatively, think of it as the total number of subcommittees minus those that are all Republicans or all Democrats.

Total subcommittees: 2002.

All Republicans: C(8,5) = 56.

All Democrats: C(6,5) = 6.

So 2002 - 56 - 6 = 1940.

Yes.

Alternatively, if I think of the possible splits:

1R and 4D: 8 * 15 = 120

2R and 3D: 28 * 20 = 560

3R and 2D: 56 * 15 = 840

4R and 1D: 70 * 6 = 420

Total: 120 + 560 + 840 + 420 = 1940.

Same answer.

Therefore, I think I can be confident that 1940 is the correct answer.

But just to make sure, let me check if there's any possibility that I miscalculated any of the combination numbers.

For example, C(8,1) is 8, correct. C(6,4) is 15. 8*15=120. Correct.

C(8,2) is 28. C(6,3) is 20. 28*20=560. Correct.

C(8,3)=56. C(6,2)=15. 56*15=840. Correct.

C(8,4)=70. C(6,1)=6. 70*6=420. Correct.

Adding them up: 120 + 560 is 680. 680 + 840 is 1520. 1520 + 420 is 1940. Correct.

Alternatively, if I think of C(8,5) = 56. Let me compute that again.

C(8,5) = 8! / (5! * 3!) = (8*7*6)/6 = 56. Correct.

C(6,5)=6. Correct.

Total subcommittees C(14,5)=2002. Correct.

So, all numbers check out.

Therefore, I think the answer is 1940.

**Final Answer**
The number of ways is \boxed{1940}.

To determine the number of ways to form a subcommittee of 5 members from a Senate committee consisting of 8 Republicans and 6 Democrats, with the condition that the subcommittee must include **at least one member from each party**, we can approach the problem using **combinatorics** and **inclusion-exclusion**.

---

### Step 1: Total Number of Subcommittees (Without Restrictions)

We are choosing 5 members from a total of 14 (8 Republicans + 6 Democrats). The total number of ways to choose 5 members from 14 is:

$$
\binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002
$$

---

### Step 2: Subtract Invalid Subcommittees

We now subtract the number of subcommittees that **do not meet the requirement** of having at least one member from each party. These are the subcommittees that are **all Republicans** or **all Democrats**.

- **All Republicans**: Choose 5 Republicans from 8:
  $$
  \binom{8}{5} = \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
  $$

- **All Democrats**: Choose 5 Democrats from 6:
  $$
  \binom{6}{5} = \binom{6}{1} = 6
  $$

So, the number of **invalid** subcommittees is:
$$
56 + 6 = 62
$$

---

### Step 3: Valid Subcommittees (At Least One from Each Party)

Now, subtract the invalid subcommittees from the total:
$$
2002 - 62 = 1940
$$

---

### Step 4: Verification via Case Analysis

We can also verify this by considering all valid combinations of Republicans and Democrats that sum to 5 members, with at least one from each party. The valid splits are:

- 1R + 4D: $ \binom{8}{1} \times \binom{6}{4} = 8 \times 15 = 120 $
- 2R + 3D: $ \binom{8}{2} \times \binom{6}{3} = 28 \times 20 = 560 $
- 3R + 2D: $ \binom{8}{3} \times \binom{6}{2} = 56 \times 15 = 840 $
- 4R + 1D: $ \binom{8}{4} \times \binom{6}{1} = 70 \times 6 = 420 $

Adding these:
$$
120 + 560 + 840 + 420 = 1940
$$

This confirms the result.

---

### Final Answer

$$
\boxed{1940}
$$

Solution: There are a total of $\binom{14}{5}=2002$ ways of selecting a subcommittee of 5 with no restrictions on the membership. Of these committees, the only ones that will violate the given condition are the ones that consist entirely of Republicans or entirely of Democrats. There are $\binom{8}{5}=56$ possible subcommittees that have all 5 members selected from among the 8 Republicans and $\binom{6}{5}=6$ possible subcommittees that have all 5 members selected from among the 6 Democrats. Subtracting the number of subcommittees that don't work from the total number of possible subcommittees gives us our answer: $2002-56-6=\boxed{1940}$.


Explanation:
Okay, so I need to simplify this expression: the square root of a fraction, where the numerator is the sum of three terms: 5 over the square root of 80, plus the square root of 845 over 9, plus the square root of 45. And all of that is divided by the square root of 5. Then take the square root of the whole thing. Hmm, that sounds complicated, but maybe if I break it down step by step, it will make more sense.

First, let me write down the expression again to make sure I have it right:

√[( (5/√80) + (√845)/9 + √45 ) / √5]

Alright, so the main structure is square root of [ (something) divided by √5 ]. Maybe I can simplify the numerator first before dealing with the denominator. Let me focus on simplifying each term in the numerator individually.

Starting with the first term: 5 divided by √80. I know that square roots can often be simplified by factoring out perfect squares. Let me see if I can simplify √80. 80 factors into 16 * 5, right? Because 16 is a perfect square. So √80 = √(16*5) = √16 * √5 = 4√5. So then 5 divided by √80 is 5 divided by 4√5. Let me write that as 5/(4√5). Maybe I can simplify this further by rationalizing the denominator? Let me try that.

Multiply numerator and denominator by √5: [5 * √5] / [4√5 * √5] = [5√5] / [4*5] = [5√5]/20 = √5/4. Oh, that's simpler! So the first term simplifies to √5/4. Okay, that's good progress.

Now moving on to the second term: √845 divided by 9. Let me simplify √845. Let me factor 845. Hmm, 845 divided by 5 is 169. Oh, 169 is 13 squared. So √845 = √(5*13²) = √(13² * 5) = 13√5. Therefore, √845 / 9 is 13√5 / 9. That seems straightforward. So the second term is 13√5 over 9.

Third term is √45. Let me simplify that. 45 is 9*5, so √45 = √(9*5) = √9 * √5 = 3√5. So the third term is 3√5.

So now, putting all the simplified terms back into the numerator:

First term: √5/4

Second term: 13√5/9

Third term: 3√5

So the numerator is √5/4 + 13√5/9 + 3√5. Now, since all terms have √5, maybe I can factor that out? Let me try that.

Let me write each term with √5:

√5*(1/4 + 13/9 + 3). Wait, but the third term is 3√5, which is 3√5/1. So to combine them, I need a common denominator for the fractions. Let me compute the sum inside the parentheses.

First, let me compute 1/4 + 13/9 + 3. Let me convert all terms to fractions with denominator... Let me see. The denominators are 4, 9, and 1. The least common denominator of 4 and 9 is 36. So let me convert each term:

1/4 = 9/36

13/9 = (13*4)/36 = 52/36

3 is 3/1 = (3*36)/36 = 108/36

So adding them up: 9/36 + 52/36 + 108/36 = (9 + 52 + 108)/36 = let me compute that.

9 + 52 is 61, plus 108 is 169. So total is 169/36. Therefore, the numerator simplifies to √5*(169/36). So the entire numerator is (169√5)/36.

So now, going back to the original expression, the entire numerator is (169√5)/36, and then we have to divide that by √5. So the whole expression inside the square root is [(169√5)/36] divided by √5. Let me write that as:

[(169√5)/36] / √5 = (169√5)/36 * 1/√5 = 169√5 / (36√5)

Now, √5 divided by √5 is 1, so this simplifies to 169/36. Wait, really? Let me check that again.

Yes, because √5 divided by √5 is 1, so the √5 cancels out. Therefore, the entire expression inside the square root simplifies to 169/36. Therefore, the original expression is the square root of 169/36.

Now, square root of 169 is 13, and square root of 36 is 6. Therefore, √(169/36) = 13/6. So the simplified fraction is 13/6.

Wait, let me just recap all steps to make sure I didn't make a mistake. Starting with the first term: 5 over sqrt(80). Simplified sqrt(80) to 4 sqrt(5), so 5/(4 sqrt(5)) becomes sqrt(5)/4. Then second term sqrt(845)/9. sqrt(845) is sqrt(169*5) which is 13 sqrt(5), so 13 sqrt(5)/9. Third term sqrt(45) is 3 sqrt(5). Then adding them up: sqrt(5)/4 + 13 sqrt(5)/9 + 3 sqrt(5). Factored out sqrt(5) and converted to common denominator 36. Got 1/4 as 9/36, 13/9 as 52/36, and 3 as 108/36. Added to 169/36. Then multiplied by sqrt(5) gives 169 sqrt(5)/36. Divided by sqrt(5) gives 169/36. Square root of that is 13/6. Seems correct.

But let me check if there's any step where I might have miscalculated. Let me verify the addition of 9 + 52 + 108. 9 + 52 is 61, 61 + 108 is 169. Yes, that's correct. 169 is 13 squared, so that's good. Then 169/36 under the square root. Square root of 169 is 13, square root of 36 is 6. So 13/6. So the simplified fraction is 13/6. That seems right.

But let me check if I handled the division by sqrt(5) correctly. The numerator was (169√5)/36 divided by √5. So dividing by √5 is multiplying by 1/√5. So (169√5)/36 * 1/√5 = 169/36 * (√5 / √5) = 169/36 * 1 = 169/36. Yes, that's correct. So that step is okay.

Alternatively, maybe I can think of the entire expression as:

Original expression: sqrt( [ (5/sqrt(80) + sqrt(845)/9 + sqrt(45)) / sqrt(5) ] )

Alternatively, maybe I can combine the terms differently? Let me try another approach to verify.

Alternatively, maybe I can rationalize each term first and then combine. Wait, but I think my previous steps are correct. Let me check each term again.

First term: 5 / sqrt(80). sqrt(80) is 4 sqrt(5). So 5 / (4 sqrt(5)) = (5 / 4) * (1 / sqrt(5)) = (5 / 4) * sqrt(5)/5 = sqrt(5)/4. Yes, that's correct. So that term is correct.

Second term: sqrt(845)/9. sqrt(845) is sqrt(169*5) = 13 sqrt(5). So 13 sqrt(5)/9. Correct.

Third term: sqrt(45) is 3 sqrt(5). Correct.

Adding them up: sqrt(5)/4 + 13 sqrt(5)/9 + 3 sqrt(5). Let me convert them all to decimals to check? Maybe that's overkill, but just to verify.

Let me compute each term numerically:

First term: sqrt(5)/4 ≈ 2.236 / 4 ≈ 0.559

Second term: 13 sqrt(5)/9 ≈ 13 * 2.236 / 9 ≈ 29.068 / 9 ≈ 3.2298

Third term: 3 sqrt(5) ≈ 3 * 2.236 ≈ 6.708

Adding them up: 0.559 + 3.2298 + 6.708 ≈ 0.559 + 3.2298 is 3.7888 + 6.708 ≈ 10.4968

Now, let me compute the entire numerator divided by sqrt(5):

So numerator ≈ 10.4968, divided by sqrt(5) ≈ 2.236. So 10.4968 / 2.236 ≈ let me compute that. 2.236 * 4.7 ≈ 2.236 * 4 + 2.236 * 0.7 ≈ 8.944 + 1.565 ≈ 10.509. Which is very close to 10.4968. So approximately 4.7. Then taking the square root of that would be sqrt(4.7) ≈ 2.167. Now, 13/6 is approximately 2.166..., which matches. So that numerical check seems to confirm that the answer is 13/6. So that gives me more confidence.

Alternatively, if I compute 169/36, that is approximately 4.694, whose square root is approx 2.166, which is 13/6. So that's correct.

Therefore, I think my answer is correct. So the simplified fraction is 13/6.

**Final Answer**
\boxed{\dfrac{13}{6}}

To simplify the expression:

$$
\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt{5}}}
$$

we proceed step by step, simplifying each term and combining them appropriately.

---

### Step 1: Simplify Each Term in the Numerator

#### First Term: $\dfrac{5}{\sqrt{80}}$

We simplify $\sqrt{80}$:

$$
\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}
$$

So:

$$
\dfrac{5}{\sqrt{80}} = \dfrac{5}{4\sqrt{5}} = \dfrac{5}{4\sqrt{5}} \cdot \dfrac{\sqrt{5}}{\sqrt{5}} = \dfrac{5\sqrt{5}}{20} = \dfrac{\sqrt{5}}{4}
$$

#### Second Term: $\dfrac{\sqrt{845}}{9}$

We simplify $\sqrt{845}$:

$$
\sqrt{845} = \sqrt{169 \cdot 5} = 13\sqrt{5}
$$

So:

$$
\dfrac{\sqrt{845}}{9} = \dfrac{13\sqrt{5}}{9}
$$

#### Third Term: $\sqrt{45}$

$$
\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}
$$

---

### Step 2: Combine the Simplified Terms

Now, the numerator becomes:

$$
\dfrac{\sqrt{5}}{4} + \dfrac{13\sqrt{5}}{9} + 3\sqrt{5}
$$

Factor out $\sqrt{5}$:

$$
\sqrt{5} \left( \dfrac{1}{4} + \dfrac{13}{9} + 3 \right)
$$

We now compute the sum inside the parentheses. To do this, find a common denominator. The least common denominator of 4, 9, and 1 is 36.

Convert each term:

- $\dfrac{1}{4} = \dfrac{9}{36}$
- $\dfrac{13}{9} = \dfrac{52}{36}$
- $3 = \dfrac{108}{36}$

Add them:

$$
\dfrac{9}{36} + \dfrac{52}{36} + \dfrac{108}{36} = \dfrac{169}{36}
$$

So the numerator becomes:

$$
\sqrt{5} \cdot \dfrac{169}{36} = \dfrac{169\sqrt{5}}{36}
$$

---

### Step 3: Divide by $\sqrt{5}$

Now, the entire expression inside the square root is:

$$
\dfrac{\dfrac{169\sqrt{5}}{36}}{\sqrt{5}} = \dfrac{169\sqrt{5}}{36\sqrt{5}} = \dfrac{169}{36}
$$

(Since $\sqrt{5}/\sqrt{5} = 1$)

---

### Step 4: Take the Square Root

$$
\sqrt{\dfrac{169}{36}} = \dfrac{\sqrt{169}}{\sqrt{36}} = \dfrac{13}{6}
$$

---

### Final Answer

$$
\boxed{\dfrac{13}{6}}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: the square root of a fraction, where the numerator is the sum of three terms: 5 over the square root of 80, plus the square root of 845 over 9, plus the square root of 45. And all of that is divided by the square root of 5. Then take the square root of the whole thing. Hmm, that sounds complicated, but maybe if I break it down step by step, it will make more sense.

First, let me write down the expression again to make sure I have it right:

√[( (5/√80) + (√845)/9 + √45 ) / √5]

Alright, so the main structure is square root of [ (something) divided by √5 ]. Maybe I can simplify the numerator first before dealing with the denominator. Let me focus on simplifying each term in the numerator individually.

Starting with the first term: 5 divided by √80. I know that square roots can often be simplified by factoring out perfect squares. Let me see if I can simplify √80. 80 factors into 16 * 5, right? Because 16 is a perfect square. So √80 = √(16*5) = √16 * √5 = 4√5. So then 5 divided by √80 is 5 divided by 4√5. Let me write that as 5/(4√5). Maybe I can simplify this further by rationalizing the denominator? Let me try that.

Multiply numerator and denominator by √5: [5 * √5] / [4√5 * √5] = [5√5] / [4*5] = [5√5]/20 = √5/4. Oh, that's simpler! So the first term simplifies to √5/4. Okay, that's good progress.

Now moving on to the second term: √845 divided by 9. Let me simplify √845. Let me factor 845. Hmm, 845 divided by 5 is 169. Oh, 169 is 13 squared. So √845 = √(5*13²) = √(13² * 5) = 13√5. Therefore, √845 / 9 is 13√5 / 9. That seems straightforward. So the second term is 13√5 over 9.

Third term is √45. Let me simplify that. 45 is 9*5, so √45 = √(9*5) = √9 * √5 = 3√5. So the third term is 3√5.

So now, putting all the simplified terms back into the numerator:

First term: √5/4

Second term: 13√5/9

Third term: 3√5

So the numerator is √5/4 + 13√5/9 + 3√5. Now, since all terms have √5, maybe I can factor that out? Let me try that.

Let me write each term with √5:

√5*(1/4 + 13/9 + 3). Wait, but the third term is 3√5, which is 3√5/1. So to combine them, I need a common denominator for the fractions. Let me compute the sum inside the parentheses.

First, let me compute 1/4 + 13/9 + 3. Let me convert all terms to fractions with denominator... Let me see. The denominators are 4, 9, and 1. The least common denominator of 4 and 9 is 36. So let me convert each term:

1/4 = 9/36

13/9 = (13*4)/36 = 52/36

3 is 3/1 = (3*36)/36 = 108/36

So adding them up: 9/36 + 52/36 + 108/36 = (9 + 52 + 108)/36 = let me compute that.

9 + 52 is 61, plus 108 is 169. So total is 169/36. Therefore, the numerator simplifies to √5*(169/36). So the entire numerator is (169√5)/36.

So now, going back to the original expression, the entire numerator is (169√5)/36, and then we have to divide that by √5. So the whole expression inside the square root is [(169√5)/36] divided by √5. Let me write that as:

[(169√5)/36] / √5 = (169√5)/36 * 1/√5 = 169√5 / (36√5)

Now, √5 divided by √5 is 1, so this simplifies to 169/36. Wait, really? Let me check that again.

Yes, because √5 divided by √5 is 1, so the √5 cancels out. Therefore, the entire expression inside the square root simplifies to 169/36. Therefore, the original expression is the square root of 169/36.

Now, square root of 169 is 13, and square root of 36 is 6. Therefore, √(169/36) = 13/6. So the simplified fraction is 13/6.

Wait, let me just recap all steps to make sure I didn't make a mistake. Starting with the first term: 5 over sqrt(80). Simplified sqrt(80) to 4 sqrt(5), so 5/(4 sqrt(5)) becomes sqrt(5)/4. Then second term sqrt(845)/9. sqrt(845) is sqrt(169*5) which is 13 sqrt(5), so 13 sqrt(5)/9. Third term sqrt(45) is 3 sqrt(5). Then adding them up: sqrt(5)/4 + 13 sqrt(5)/9 + 3 sqrt(5). Factored out sqrt(5) and converted to common denominator 36. Got 1/4 as 9/36, 13/9 as 52/36, and 3 as 108/36. Added to 169/36. Then multiplied by sqrt(5) gives 169 sqrt(5)/36. Divided by sqrt(5) gives 169/36. Square root of that is 13/6. Seems correct.

But let me check if there's any step where I might have miscalculated. Let me verify the addition of 9 + 52 + 108. 9 + 52 is 61, 61 + 108 is 169. Yes, that's correct. 169 is 13 squared, so that's good. Then 169/36 under the square root. Square root of 169 is 13, square root of 36 is 6. So 13/6. So the simplified fraction is 13/6. That seems right.

But let me check if I handled the division by sqrt(5) correctly. The numerator was (169√5)/36 divided by √5. So dividing by √5 is multiplying by 1/√5. So (169√5)/36 * 1/√5 = 169/36 * (√5 / √5) = 169/36 * 1 = 169/36. Yes, that's correct. So that step is okay.

Alternatively, maybe I can think of the entire expression as:

Original expression: sqrt( [ (5/sqrt(80) + sqrt(845)/9 + sqrt(45)) / sqrt(5) ] )

Alternatively, maybe I can combine the terms differently? Let me try another approach to verify.

Alternatively, maybe I can rationalize each term first and then combine. Wait, but I think my previous steps are correct. Let me check each term again.

First term: 5 / sqrt(80). sqrt(80) is 4 sqrt(5). So 5 / (4 sqrt(5)) = (5 / 4) * (1 / sqrt(5)) = (5 / 4) * sqrt(5)/5 = sqrt(5)/4. Yes, that's correct. So that term is correct.

Second term: sqrt(845)/9. sqrt(845) is sqrt(169*5) = 13 sqrt(5). So 13 sqrt(5)/9. Correct.

Third term: sqrt(45) is 3 sqrt(5). Correct.

Adding them up: sqrt(5)/4 + 13 sqrt(5)/9 + 3 sqrt(5). Let me convert them all to decimals to check? Maybe that's overkill, but just to verify.

Let me compute each term numerically:

First term: sqrt(5)/4 ≈ 2.236 / 4 ≈ 0.559

Second term: 13 sqrt(5)/9 ≈ 13 * 2.236 / 9 ≈ 29.068 / 9 ≈ 3.2298

Third term: 3 sqrt(5) ≈ 3 * 2.236 ≈ 6.708

Adding them up: 0.559 + 3.2298 + 6.708 ≈ 0.559 + 3.2298 is 3.7888 + 6.708 ≈ 10.4968

Now, let me compute the entire numerator divided by sqrt(5):

So numerator ≈ 10.4968, divided by sqrt(5) ≈ 2.236. So 10.4968 / 2.236 ≈ let me compute that. 2.236 * 4.7 ≈ 2.236 * 4 + 2.236 * 0.7 ≈ 8.944 + 1.565 ≈ 10.509. Which is very close to 10.4968. So approximately 4.7. Then taking the square root of that would be sqrt(4.7) ≈ 2.167. Now, 13/6 is approximately 2.166..., which matches. So that numerical check seems to confirm that the answer is 13/6. So that gives me more confidence.

Alternatively, if I compute 169/36, that is approximately 4.694, whose square root is approx 2.166, which is 13/6. So that's correct.

Therefore, I think my answer is correct. So the simplified fraction is 13/6.

**Final Answer**
\boxed{\dfrac{13}{6}}

To simplify the expression:

$$
\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt{5}}}
$$

we proceed step by step, simplifying each term and combining them appropriately.

---

### Step 1: Simplify Each Term in the Numerator

#### First Term: $\dfrac{5}{\sqrt{80}}$

We simplify $\sqrt{80}$:

$$
\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}
$$

So:

$$
\dfrac{5}{\sqrt{80}} = \dfrac{5}{4\sqrt{5}} = \dfrac{5}{4\sqrt{5}} \cdot \dfrac{\sqrt{5}}{\sqrt{5}} = \dfrac{5\sqrt{5}}{20} = \dfrac{\sqrt{5}}{4}
$$

#### Second Term: $\dfrac{\sqrt{845}}{9}$

We simplify $\sqrt{845}$:

$$
\sqrt{845} = \sqrt{169 \cdot 5} = 13\sqrt{5}
$$

So:

$$
\dfrac{\sqrt{845}}{9} = \dfrac{13\sqrt{5}}{9}
$$

#### Third Term: $\sqrt{45}$

$$
\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}
$$

---

### Step 2: Combine the Simplified Terms

Now, the numerator becomes:

$$
\dfrac{\sqrt{5}}{4} + \dfrac{13\sqrt{5}}{9} + 3\sqrt{5}
$$

Factor out $\sqrt{5}$:

$$
\sqrt{5} \left( \dfrac{1}{4} + \dfrac{13}{9} + 3 \right)
$$

We now compute the sum inside the parentheses. To do this, find a common denominator. The least common denominator of 4, 9, and 1 is 36.

Convert each term:

- $\dfrac{1}{4} = \dfrac{9}{36}$
- $\dfrac{13}{9} = \dfrac{52}{36}$
- $3 = \dfrac{108}{36}$

Add them:

$$
\dfrac{9}{36} + \dfrac{52}{36} + \dfrac{108}{36} = \dfrac{169}{36}
$$

So the numerator becomes:

$$
\sqrt{5} \cdot \dfrac{169}{36} = \dfrac{169\sqrt{5}}{36}
$$

---

### Step 3: Divide by $\sqrt{5}$

Now, the entire expression inside the square root is:

$$
\dfrac{\dfrac{169\sqrt{5}}{36}}{\sqrt{5}} = \dfrac{169\sqrt{5}}{36\sqrt{5}} = \dfrac{169}{36}
$$

(Since $\sqrt{5}/\sqrt{5} = 1$)

---

### Step 4: Take the Square Root

$$
\sqrt{\dfrac{169}{36}} = \dfrac{\sqrt{169}}{\sqrt{36}} = \dfrac{13}{6}
$$

---

### Final Answer

$$
\boxed{\dfrac{13}{6}}
$$

Solution:
First, we'll divide $\sqrt{5}$ into each term in the numerator of the fraction inside the big radical: $$\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt5}}=
\sqrt{\frac{5}{\sqrt{80}\cdot\sqrt{5}} + \frac{\sqrt{845}}{9\sqrt{5}} + \frac{\sqrt{45}}{\sqrt{5}}}.
$$Let's attack each fraction within the square root separately. First, $$\dfrac{5}{\sqrt{80}\cdot\sqrt5}=\dfrac{5}{\sqrt{400}}=\dfrac{5}{20}=\dfrac{1}{4}.$$The second one is trickier: $$\dfrac{\sqrt{845}}{9\sqrt5}=\dfrac{\sqrt{169}}{9}=\dfrac{13}{9}.$$Finally, $\dfrac{\sqrt{45}}{\sqrt5}=\sqrt9=3$. Adding these together, we get $$\sqrt{\dfrac{1}{4}+\dfrac{13}{9}+3}=\sqrt{\dfrac{9+52+108}{36}}=\sqrt{\dfrac{169}{36}}=\boxed{\frac{13}{6}}.$$