Question

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

Explanation:

Step1: Recall the exponent rule for rational exponents

The formula for a rational exponent \(a^{\frac{m}{n}}\) is \(\sqrt[n]{a^m}\) (or \((\sqrt[n]{a})^m\)). Here, we have \(-32^{\frac{3}{5}}\), so we can rewrite the base \(32\) as a power of \(2\) since \(32 = 2^5\).

Step2: Substitute \(32 = 2^5\) into the expression

We get \(-(2^5)^{\frac{3}{5}}\).

Step3: Apply the power - of - a - power rule \((a^m)^n=a^{mn}\)

For \((2^5)^{\frac{3}{5}}\), using the rule \((a^m)^n=a^{m\times n}\), we have \(2^{5\times\frac{3}{5}}\). The \(5\) in the numerator and denominator cancels out, leaving \(2^3\).

Step4: Calculate \(2^3\)

We know that \(2^3=8\). Since we have a negative sign in front, the value of \(-32^{\frac{3}{5}}\) is \(-8\).

Let's also check the other options:

• Option 2: \(-\sqrt[3]{32^5}\) is equivalent to \(-32^{\frac{5}{3}}\), which is not the same as \(-32^{\frac{3}{5}}\).
• Option 3: \(\frac{1}{\sqrt[3]{32^5}}\) is equivalent to \(32^{-\frac{5}{3}}\), which is not the same as \(-32^{\frac{3}{5}}\).
• Option 4: \(\frac{1}{8}\) is positive and our result is negative, so it's incorrect.

Answer:

\(-8\) (the first option)