Question

express $27^{\frac{5}{3}}$ 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, \(a = 27\), \(m = 5\), \(n = 3\). Also, note that \(27=3^3\).

So we can rewrite \(27^{\frac{5}{3}}\) as \((3^3)^{\frac{5}{3}}\).

Step2: Use the exponent rule \((a^m)^n=a^{m\times n}\). For \((3^3)^{\frac{5}{3}}\), multiply the exponents: \(3\times\frac{5}{3}=5\). So \((3^3)^{\frac{5}{3}} = 3^5\).

Step3: Calculate \(3^5\). \(3^5=3\times3\times3\times3\times3 = 243\).

Alternatively, using the first form of the exponent rule: \(27^{\frac{5}{3}}=\sqrt[3]{27^5}\). Since \(27 = 3^3\), then \(27^5=(3^3)^5=3^{15}\). So \(\sqrt[3]{3^{15}}=3^{\frac{15}{3}} = 3^5=243\).

Answer:

\(243\)