The problem asks us to evaluate the expression $\frac{1}{2} \cdot 0.3^{1/3} \times 0.3^{1/3} \times 0.3^{1/3}$.

1. Problem Description

The problem asks us to evaluate the expression $\frac{1}{2} \cdot 0.3^{1/3} \times 0.3^{1/3} \times 0.3^{1/3}$.

2. Solution Steps

We have the expression $\frac{1}{2} \cdot 0.3^{1/3} \times 0.3^{1/3} \times 0.3^{1/3}$.

We can rewrite the expression as $\frac{1}{2} \cdot (0.3^{1/3})^3$.

Using the property of exponents $a^m \times a^n = a^{m+n}$, or $(a^m)^n = a^{m \times n}$, we have:

$$(0.3^{1/3})^3 = 0.3^{(1/3) \times 3} = 0.3^1 = 0.3$$

Therefore, the expression becomes:

$\frac{1}{2} \cdot 0.3 = \frac{0.3}{2} = 0.15$.

3. Final Answer

The final answer is 0.15.