Question

question
express $32^{\frac{3}{5}}$ in simplest radical form.

Explanation:

Step1: Recall the exponent rule \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\) or \(a^{\frac{m}{n}}=(\sqrt[n]{a})^m\). Here, \(a = 32\), \(m = 3\), \(n = 5\). Also, note that \(32=2^5\).

So we can rewrite \(32^{\frac{3}{5}}\) as \((2^5)^{\frac{3}{5}}\).

Step2: Use the exponent rule \((a^m)^n=a^{mn}\). Applying this to \((2^5)^{\frac{3}{5}}\), we get \(2^{5\times\frac{3}{5}}\).

Simplify the exponent: \(5\times\frac{3}{5}=3\). So we have \(2^3\).

Step3: Calculate \(2^3\). \(2^3 = 8\).

Answer:

\(8\)