The problem is to solve the equation $(1-a)^3 = \frac{1}{2}$ for $a$.

1. Problem Description

The problem is to solve the equation $(1-a)^3 = \frac{1}{2}$ for $a$.

2. Solution Steps

First, take the cube root of both sides of the equation:

$$(1-a)^3 = \frac{1}{2}$$

$$\sqrt[3]{(1-a)^3} = \sqrt[3]{\frac{1}{2}}$$

$$1-a = \frac{1}{\sqrt[3]{2}}$$

Next, isolate $a$ by subtracting 1 from both sides and multiplying by $-1$:

$$-a = \frac{1}{\sqrt[3]{2}} - 1$$

$$a = 1 - \frac{1}{\sqrt[3]{2}}$$

We can rationalize the denominator by multiplying by $\frac{\sqrt[3]{4}}{\sqrt[3]{4}}$ to get:

$$a = 1 - \frac{\sqrt[3]{4}}{2}$$

3. Final Answer

$a = 1 - \frac{1}{\sqrt[3]{2}} = 1 - \frac{\sqrt[3]{4}}{2}$