Question

express \\(\left(8^{\frac{1}{3}}\
ight)^5\\) in simplest radical form.

Explanation:

Step1: Use exponent power rule

When raising a power to a power, we multiply the exponents. The formula is \((a^{m})^{n}=a^{m\times n}\). Here, \(a = 8\), \(m=\frac{1}{3}\), and \(n = 5\). So we have \(8^{\frac{1}{3}\times5}=8^{\frac{5}{3}}\).

Step2: Rewrite the exponent as a sum

We can rewrite \(\frac{5}{3}\) as \(1+\frac{2}{3}\), so \(8^{\frac{5}{3}}=8^{1 + \frac{2}{3}}\).

Step3: Use exponent product rule

The exponent product rule is \(a^{m + n}=a^{m}\times a^{n}\). So \(8^{1+\frac{2}{3}}=8^{1}\times8^{\frac{2}{3}}\).

Step4: Simplify \(8^{\frac{2}{3}}\)

We know that \(8 = 2^{3}\), so substitute that in: \((2^{3})^{\frac{2}{3}}\). Then use the power - of - a - power rule \((a^{m})^{n}=a^{m\times n}\), so \((2^{3})^{\frac{2}{3}}=2^{3\times\frac{2}{3}}=2^{2}=4\).

Step5: Calculate the final result

Since \(8^{1}=8\) and \(8^{\frac{2}{3}} = 4\), then \(8\times4 = 32\). Also, we can think of \(8^{\frac{5}{3}}\) as \(\sqrt[3]{8^{5}}\). Since \(8 = 2^{3}\), \(8^{5}=(2^{3})^{5}=2^{15}\), and \(\sqrt[3]{2^{15}}=2^{5}=32\) (because \(\sqrt[3]{a^{m}}=a^{\frac{m}{3}}\), so when \(m = 15\), \(\frac{m}{3}=5\)).

Answer:

\(32\)