To evaluate the equation:

$\frac{x^2}{x^{\frac{1}{2}}} = x^a$

Step 1: Simplify the left-hand side

Using the rule of exponents: $\frac{x^m}{x^n} = x^{m-n}$

Here, $m = 2$ and $n = \frac{1}{2}$. So: $\frac{x^2}{x^{\frac{1}{2}}} = x^{2 - \frac{1}{2}}$

Step 2: Simplify the exponent

$2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}$

Thus, the left-hand side becomes: $x^{\frac{3}{2}}$

Step 3: Equating powers of $x$

Since the left-hand side equals $x^a$, we equate the exponents: $a = \frac{3}{2}$

Final Answer:

$a = \frac{3}{2}$

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Would you like a detailed explanation of exponent rules? Here are some related questions:

1. What are the rules for multiplying and dividing powers with the same base?
2. How can exponents be applied to simplify fractions involving variables?
3. What does it mean when the exponent is a fraction?
4. Can you explain how to convert fractional exponents into radicals?
5. How would this change if $x$ were replaced with a different variable?

Tip: When working with fractions and exponents, always simplify the powers first to avoid errors!