Which of the following expressions is equivalent to \( (-8x^2)^{\frac{2}{3}} \)

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If we seperate the items inside the parenthesis:

$$ ( -1 \cdot 8 \cdot x \cdot x)^{\frac{2}{3}} $$

We can take care of the 2 exponent first by doubling the terms inside the parenthesis.

$$ ( -1 \cdot 8 \cdot x \cdot x)^{\colorbox{aqua}{$2$}\cdot \frac{1}{3}} $$ $$ ( -1 \cdot -1 \cdot 8 \cdot 8 \cdot x \cdot x \cdot x \cdot x)^{\frac{1}{3}} $$

The negative values cancel each other out. The current exponent is equivalent to a cube root.

$$ \sqrt[3]{8\cdot 8\cdot x\cdot x \cdot x \cdot x} $$

We can take out a \(x\) since there are three of them, but we should obtain the prime factors of \(8\) to see if anything else can be simplified.

$$ \sqrt[3]{\colorbox{aqua}{$2 \cdot 2 \cdot 2$} \cdot \colorbox{aqua}{$2 \cdot 2 \cdot 2$} \cdot \colorbox{aqua}{$x \cdot x \cdot x$} \cdot x} $$ $$ = 2\cdot 2\cdot x \cdot \sqrt[3]{x} $$ $$ = 4x \sqrt[3]{x} $$