To simplify the expression:

$\frac{x^{3/4}}{x^{5/4} \cdot x^{-2}}$

Step 1: Combine the terms in the denominator

The denominator consists of $x^{5/4} \cdot x^{-2}$. To simplify this, we apply the rule of exponents:

$x^{a} \cdot x^{b} = x^{a+b}$

Thus:

$x^{5/4} \cdot x^{-2} = x^{5/4 - 2} = x^{5/4 - 8/4} = x^{-3/4}$

Step 2: Apply the rule of division for exponents

Now the expression becomes:

$\frac{x^{3/4}}{x^{-3/4}}$

When dividing terms with the same base, subtract the exponents:

$x^{3/4 - (-3/4)} = x^{3/4 + 3/4} = x^{6/4} = x^{3/2}$

Final Answer:

$x^{3/2}$

Let me know if you'd like more details or explanations.

Here are some related questions:

1. How do you apply the rule of exponents when multiplying or dividing?
2. What happens when the exponent is negative?
3. How do fractional exponents relate to roots?
4. Can you convert $x^{3/2}$ to a radical expression?
5. What is the general rule for simplifying expressions involving multiple exponents?

Tip: Always combine like terms and simplify step by step, especially with fractional exponents.