The problem asks to express $(\sqrt[7]{4x})^3$ with a rational exponent. We are given that $x$ represents a positive real number.

1. Problem Description

The problem asks to express $(\sqrt[7]{4x})^3$ with a rational exponent. We are given that $x$ represents a positive real number.

2. Solution Steps

We need to express the radical as a rational exponent and then simplify the expression.

First, we can write the radical as a fractional exponent. Recall that $\sqrt[n]{a} = a^{\frac{1}{n}}$. Therefore, $\sqrt[7]{4x} = (4x)^{\frac{1}{7}}$.

Now we have $((4x)^{\frac{1}{7}})^3$. We use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.

So, $((4x)^{\frac{1}{7}})^3 = (4x)^{\frac{1}{7} \cdot 3} = (4x)^{\frac{3}{7}}$.

3. Final Answer

$(4x)^{\frac{3}{7}}$