Let x and y be the positive real numbers such that `log_x y=-(1)/(4)`, then value of the expression `log_x(xy^5)-log_y""((x^(2))/(sqrt(y)))` is `(p)/(q)` then q is _____________. { where p , q `in N` and are coprime to each other }

Answer

To solve the problem, we need to evaluate the expression given the condition \( \log_x y = -\frac{1}{4} \). Let's break it down step by step.

### Step 1: Rewrite the logarithmic condition
Given:
\[ \log_x y = -\frac{1}{4} \]
This can be rewritten using the change of base formula:
\[ \frac{\log y}{\log x} = -\frac{1}{4} \]
From this, we can express \( \log y \) in terms of \( \log x \):
\[ \log y = -\frac{1}{4} \log x \]

### Step 2: Substitute \( \log y \) in the expression
We need to evaluate:
\[ \log_x(xy^5) - \log_y\left(\frac{x^2}{\sqrt{y}}\right) \]
First, let's break down \( \log_x(xy^5) \):
\[ \log_x(xy^5) = \log_x x + \log_x y^5 = 1 + 5 \log_x y \]
Substituting \( \log_x y = -\frac{1}{4} \):
\[ \log_x(xy^5) = 1 + 5\left(-\frac{1}{4}\right) = 1 - \frac{5}{4} = 1 - 1.25 = -0.25 = -\frac{1}{4} \]

### Step 3: Evaluate \( \log_y\left(\frac{x^2}{\sqrt{y}}\right) \)
Now, we evaluate:
\[ \log_y\left(\frac{x^2}{\sqrt{y}}\right) = \log_y x^2 - \log_y \sqrt{y} \]
Calculating each term:
\[ \log_y x^2 = 2 \log_y x \]
Using the change of base formula:
\[ \log_y x = \frac{1}{\log_x y} = -4 \quad (\text{since } \log_x y = -\frac{1}{4}) \]
Thus:
\[ \log_y x^2 = 2(-4) = -8 \]
Now, for \( \log_y \sqrt{y} \):
\[ \log_y \sqrt{y} = \log_y y^{1/2} = \frac{1}{2} \]
Putting it all together:
\[ \log_y\left(\frac{x^2}{\sqrt{y}}\right) = -8 - \frac{1}{2} = -8.5 = -\frac{17}{2} \]

### Step 4: Combine the results
Now we can combine the results:
\[ \log_x(xy^5) - \log_y\left(\frac{x^2}{\sqrt{y}}\right) = -\frac{1}{4} - \left(-\frac{17}{2}\right) \]
This simplifies to:
\[ -\frac{1}{4} + \frac{17}{2} = -\frac{1}{4} + \frac{34}{4} = \frac{33}{4} \]

### Step 5: Identify \( p \) and \( q \)
The expression evaluates to:
\[ \frac{p}{q} = \frac{33}{4} \]
Here, \( p = 33 \) and \( q = 4 \).

### Final Answer
Since \( p \) and \( q \) are coprime, the value of \( q \) is:
\[ \boxed{4} \]