Question

which expression is equivalent to \\(-32^{\frac{3}{5}}\\)?\
\\(\circ\\) \\(-8\\)\
\\(\circ\\) \\(-\sqrt3{32^5}\\)\
\\(\circ\\) \\(\frac{1}{\sqrt3{32^5}}\\)\
\\(\circ\\) \\(\frac{1}{8}\\)

Explanation:

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

For \(-32^{\frac{3}{5}}\), apply the rule: \(32^{\frac{3}{5}}=\sqrt[5]{32^3}\). But also, \(32 = 2^5\), so substitute \(32\) with \(2^5\): \((2^5)^{\frac{3}{5}}\).

Step2: Simplify \((2^5)^{\frac{3}{5}}\)

Using exponent rule \((a^m)^n=a^{mn}\), we get \(2^{5\times\frac{3}{5}} = 2^3=8\). So \(-32^{\frac{3}{5}}=-8\).

Answer:

-8 (corresponding to the first option: \(-8\))