Question

express $27^{\frac{5}{3}}$ in simplest radical form.

Explanation:

Step1: Recall the exponent rule

For a number \(a^{\frac{m}{n}}\), it can be rewritten as \(\sqrt[n]{a^m}\) or \((\sqrt[n]{a})^m\). Here, \(a = 27\), \(m=5\), \(n = 3\). Also, we know that \(27=3^3\).

Step2: Substitute \(a = 3^3\) into the expression

So \(27^{\frac{5}{3}}=(3^3)^{\frac{5}{3}}\). Using the exponent rule \((a^m)^n=a^{mn}\), we get \(3^{3\times\frac{5}{3}}\).

Step3: Simplify the exponent

\(3\times\frac{5}{3}=5\), so the expression becomes \(3^5\).

Step4: Calculate \(3^5\)

\(3^5 = 3\times3\times3\times3\times3=243\).

Answer:

\(243\)