Simplify the expression $\left(32x^{20}\right)^{-1/5}$, assuming that $x$ represents a positive real number.

1. Problem Description

Simplify the expression $\left(32x^{20}\right)^{-1/5}$, assuming that $x$ represents a positive real number.

2. Solution Steps

We want to simplify the expression $\left(32x^{20}\right)^{-1/5}$. We can use the property $(ab)^n = a^n b^n$. Also, $(a^m)^n = a^{mn}$.

$\left(32x^{20}\right)^{-1/5} = 32^{-1/5} (x^{20})^{-1/5}$

Now, $32 = 2^5$.

$32^{-1/5} = (2^5)^{-1/5} = 2^{5(-1/5)} = 2^{-1} = \frac{1}{2}$.

$(x^{20})^{-1/5} = x^{20(-1/5)} = x^{-4} = \frac{1}{x^4}$.

Therefore,

$\left(32x^{20}\right)^{-1/5} = \frac{1}{2} \cdot \frac{1}{x^4} = \frac{1}{2x^4}$.

3. Final Answer

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