The problem is to solve for $a$ in the equation $(1 - a^5) = \frac{1}{2}$.

1. Problem Description

The problem is to solve for $a$ in the equation $(1 - a^5) = \frac{1}{2}$.

2. Solution Steps

We want to isolate $a^5$ on one side of the equation.

First, subtract 1 from both sides of the equation:

$1 - a^5 - 1 = \frac{1}{2} - 1$

$-a^5 = -\frac{1}{2}$

Now, multiply both sides by $-1$:

$a^5 = \frac{1}{2}$

To find $a$, we take the fifth root of both sides:

$a = \sqrt[5]{\frac{1}{2}}$

$a = (\frac{1}{2})^{\frac{1}{5}}$

$a = \frac{1^{\frac{1}{5}}}{2^{\frac{1}{5}}}$

$a = \frac{1}{2^{\frac{1}{5}}}$

3. Final Answer

$a = \frac{1}{2^{\frac{1}{5}}}$