Question

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

Explanation:

Step1: Recall exponent rule

The rule for \(a^{\frac{m}{n}}\) is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (or \((\sqrt[n]{a})^{m}\)). Also, for negative exponents or negative signs, we consider the sign's effect. Given \(-32^{\frac{3}{5}}\), first handle the exponent.

Step2: Rewrite \(32\) as a power

We know that \(32 = 2^{5}\), so \(32^{\frac{3}{5}}=(2^{5})^{\frac{3}{5}}\).

Step3: Simplify the exponent

Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), we have \((2^{5})^{\frac{3}{5}} = 2^{5\times\frac{3}{5}}=2^{3}=8\). But we have a negative sign in front, so \(-32^{\frac{3}{5}}=-8\)? Wait, no, wait. Wait, maybe I misread. Wait, the original expression is \(-32^{\frac{3}{5}}\) or is it \((-32)^{\frac{3}{5}}\)? Wait, the problem says \(-32^{\frac{3}{5}}\). Wait, but let's check the options. Wait, the options have \(-\sqrt[5]{32^{3}}\)? Wait, no, the options: first option is \(-8\), second is \(-\sqrt[3]{325}\) (maybe a typo, should be \(32^{3}\)? Wait, no, 325? Wait, maybe it's \(32\) with exponent 3? Wait, maybe the original problem is \(-32^{\frac{3}{5}}\), and let's recall that \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), so \(32^{\frac{3}{5}}=\sqrt[5]{32^{3}}\), and with the negative sign, it's \(-\sqrt[5]{32^{3}}\)? Wait, but \(32 = 2^{5}\), so \(32^{3}=(2^{5})^{3}=2^{15}\), and \(\sqrt[5]{2^{15}} = 2^{3}=8\), so \(-\sqrt[5]{32^{3}}=-8\). Wait, but the first option is \(-8\). Wait, maybe the options have a typo, but let's check again. Wait, the expression is \(-32^{\frac{3}{5}}\). Let's compute \(32^{\frac{3}{5}}\): since \(32 = 2^{5}\), then \(32^{\frac{3}{5}}=(2^{5})^{\frac{3}{5}}=2^{3}=8\), so \(-32^{\frac{3}{5}}=-8\). So the first option is \(-8\), which should be the answer. Wait, but let's check the options again. The first option is \(-8\), so that's the equivalent expression.

Answer:

A. \(-8\)