Question from March 2018 SAT: section 3 #15
Which of the following expressions is equivalent to (-4x^3)^2/3 ?

A. -2x^3 * ∛2
B. -x^3 * ∛16
C. 2x^2 * ∛2
D. 2x^2 * ∛16

I like this question. For fractional exponents, remember that the top number is the power and the bottom is the root. Also remember that when you’re applying powers to an expression like this, you have to apply the power to all parts being multiplied. So what I’m going to do first below is break the –4 off, and then apply the exponents.

    \begin{align*}&~~~\left(-4x^3\right)^\frac{2}{3}\\&=\left(-4\right)^\frac{2}{3}\left(x^3\right)^\frac{2}{3}\\&=\left(16\right)^\frac{1}{3}\left(x^6\right)^\frac{1}{3}\\&=\sqrt[3]{16}\left(x^2\right)\end{align*}

Unfortunately, that’s not an answer choice. It is good enough to eliminate choices A and B, though, because there’s nothing else we can do to the x term. It’s also good enough to eliminate D if we’re in a hurry because D makes a 2 magically appear without changing anything else in our simplified expression. Since we’re not in a rush, though, let’s manipulate what we’ve got into choice C just to be sure.

\begin{align*}&~~~~\sqrt[3]{16}\left(x^2\right)\\&=\sqrt[3]{(2)(2)(2)(2)}\left(x^2\right)\\&=\sqrt[3]{\left(2^3\right)(2)}\left(x^2\right)\\&=\sqrt[3]{\left(2^3\right)}\sqrt[3]{2}\left(x^2\right)\\&=2x^2\cdot \sqrt[3]{2}\end{ailgn*}