Question

47 rewrite $2^{\frac{4}{3}}$ in simplest radical form.

Explanation:

Step1: Recall the exponent rule \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)

Here, \(a = 2\), \(m=4\), \(n = 3\), so \(2^{\frac{4}{3}}=\sqrt[3]{2^4}\)

Step2: Simplify the exponent in the radical

Calculate \(2^4=16\), so we have \(\sqrt[3]{16}\)

Step3: Factor 16 to simplify the radical

Since \(16 = 8\times2\) and \(\sqrt[3]{8}=2\), we can rewrite \(\sqrt[3]{16}\) as \(\sqrt[3]{8\times2}\)
Using the property \(\sqrt[n]{ab}=\sqrt[n]{a}\times\sqrt[n]{b}\) (for \(a,b\geq0\)), we get \(\sqrt[3]{8}\times\sqrt[3]{2}=2\sqrt[3]{2}\)

Answer:

\(2\sqrt[3]{2}\)