Simplify the expression $(\frac{2^5}{x^2})^4$.

1. Problem Description

Simplify the expression $(\frac{2^5}{x^2})^4$.

2. Solution Steps

We need to simplify the given expression $(\frac{2^5}{x^2})^4$.

We can distribute the exponent 4 to both the numerator and the denominator.

$(\frac{2^5}{x^2})^4 = \frac{(2^5)^4}{(x^2)^4}$

Now, we use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.

$(2^5)^4 = 2^{5 \cdot 4} = 2^{20}$

$(x^2)^4 = x^{2 \cdot 4} = x^8$

Therefore,

$\frac{(2^5)^4}{(x^2)^4} = \frac{2^{20}}{x^8}$

3. Final Answer

$\frac{2^{20}}{x^8}$