Question

which expression is equivalent to \\(-32^{\frac{3}{5}}\\)?
options:
-8
\\(-\sqrt5{32^3}\\)
\\(\frac{1}{\sqrt5{32^3}}\\)
\\(\frac{1}{16}\\)

Explanation:

Step1: Recall exponent - root relation

The formula for converting a rational exponent to a root is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (when \(a\geq0\) for even roots, and for odd roots, \(a\) can be any real number). Also, \(-a^{\frac{m}{n}}=-\sqrt[n]{a^{m}}\) and \(a^{-\frac{m}{n}}=\frac{1}{a^{\frac{m}{n}}}=\frac{1}{\sqrt[n]{a^{m}}}\)

We are given the expression \(- 32^{\frac{3}{5}}\)

Step2: Apply the exponent - root formula

For \(a = 32\), \(m = 3\), \(n=5\), using the formula \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), we have \(32^{\frac{3}{5}}=\sqrt[5]{32^{3}}\)

Since we have a negative sign in front, \(-32^{\frac{3}{5}}=-\sqrt[5]{32^{3}}\) (assuming the original expression is \(-32^{\frac{3}{5}}\), and from the options, one of the options is \(-\sqrt[5]{32^{3}}\) (matching the second option in the given choices, considering the notation in the image)

Answer:

The expression \(-\sqrt[5]{32^{3}}\) (the second option from the left, with the negative sign and the fifth - root of \(32^{3}\))