The problem is to simplify the expression $\frac{\frac{1}{4x^3}}{\frac{1}{x^{\frac{1}{2}}}}$ and express the answer with positive exponents.

1. Problem Description

The problem is to simplify the expression $\frac{\frac{1}{4x^3}}{\frac{1}{x^{\frac{1}{2}}}}$ and express the answer with positive exponents.

2. Solution Steps

We have the expression

$\frac{\frac{1}{4x^3}}{\frac{1}{x^{\frac{1}{2}}}}$

We can rewrite this as a division problem:

$\frac{1}{4x^3} \div \frac{1}{x^{\frac{1}{2}}}$

Dividing by a fraction is the same as multiplying by its reciprocal.

So, we have:

$\frac{1}{4x^3} \times x^{\frac{1}{2}}$

This simplifies to:

$\frac{x^{\frac{1}{2}}}{4x^3}$

Now we can simplify further using the rule for dividing exponents with the same base, which is $\frac{x^a}{x^b} = x^{a-b}$.

So,

$\frac{x^{\frac{1}{2}}}{4x^3} = \frac{1}{4} \cdot \frac{x^{\frac{1}{2}}}{x^3} = \frac{1}{4} x^{\frac{1}{2} - 3}$

To subtract the exponents, we need a common denominator. So $3 = \frac{6}{2}$.

$\frac{1}{2} - 3 = \frac{1}{2} - \frac{6}{2} = \frac{1-6}{2} = \frac{-5}{2}$

So we have:

$\frac{1}{4} x^{\frac{-5}{2}}$

Since we need positive exponents, we can rewrite this as:

$\frac{1}{4x^{\frac{5}{2}}}$

3. Final Answer

$\frac{1}{4x^{\frac{5}{2}}}$