Question

which expression is equivalent to $-32^{\frac{3}{5}}$? $-sqrt3{32^{5}}$, $\frac{1}{8}$, $\frac{1}{sqrt3{32^{5}}}$, $-8$

Explanation:

Step1: Recall the exponent rule

The rule for fractional exponents is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (when \(a\geq0\) for even roots, but here we have a negative sign and odd root context). So for \(-32^{\frac{3}{5}}\), we can rewrite the exponent.

Step2: Apply the exponent rule

First, \(32^{\frac{3}{5}}=\sqrt[5]{32^{3}}\)? Wait, no, wait: the general rule is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) or \((\sqrt[n]{a})^{m}\). Let's compute \(32\) as a power of \(2\), since \(2^{5} = 32\). So \(32=2^{5}\). Then \(32^{\frac{3}{5}}=(2^{5})^{\frac{3}{5}}\). Using the power of a power rule \((a^{m})^{n}=a^{mn}\), we get \(2^{5\times\frac{3}{5}}=2^{3}=8\). Then the original expression is \(-32^{\frac{3}{5}}=-8\). Let's check the options:

• First option: \(-\sqrt[3]{32^{5}}\) is not equivalent, since the root index and exponent are different.
• Second option: \(\frac{1}{8}\) is positive, but our result is \(-8\), so no.
• Third option: \(\frac{1}{\sqrt[3]{32^{5}}}\) is also positive and not related.
• Fourth option: \(-8\) matches our calculation.

Answer:

\(-8\) (the fourth option, assuming the fourth option is \(-8\))