Question

question
select the expression that is equivalent to ((2x)^{\frac{3}{2}})

answer attempt 2 out of 2
( \bigcirc \frac{1}{sqrt{(2x)^3}} ) ( \bigcirc sqrt3{(2x)^2} )
( \bigcirc sqrt{(2x)^3} ) ( \boldsymbol{\bigcirc} \frac{1}{sqrt3{(2x)^2}} )

Explanation:

Step1: Recall the exponent rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\)

For the expression \((2x)^{\frac{3}{2}}\), using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) where \(a = 2x\), \(m=3\) and \(n = 2\), we get \(\sqrt[2]{(2x)^{3}}\) which is the same as \(\sqrt{(2x)^{3}}\) since the index of a square root is 2 (and we usually don't write the index 2 for square roots).

Let's analyze the other options:

• For \(\frac{1}{\sqrt{(2x)^{3}}}\), this is \((2x)^{-\frac{3}{2}}\) (using \(a^{-n}=\frac{1}{a^{n}}\) and \(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\)), which is not equivalent to \((2x)^{\frac{3}{2}}\).
• For \(\sqrt[3]{(2x)^{2}}\), using the exponent rule, this is \((2x)^{\frac{2}{3}}\), which is not equivalent to \((2x)^{\frac{3}{2}}\).
• For \(\frac{1}{\sqrt[3]{(2x)^{2}}}\), this is \((2x)^{-\frac{2}{3}}\), which is not equivalent to \((2x)^{\frac{3}{2}}\).

Answer:

\(\boldsymbol{\sqrt{(2x)^{3}}}\) (or the option corresponding to \(\sqrt{(2x)^{3}}\))