Question

which expression represents (\\(\sqrt3{8}\\))^4 in rational exponent form?
(\\(\frac{1}{8}\\))^4
(\\(\frac{1}{8}\\))^(1/4)
8^(1/4)
8^(4/3)

Explanation:

Step1: Recall radical - exponent conversion

The n - th root of a number \(a\) can be written as \(a^{\frac{1}{n}}\). So, \(\sqrt[3]{8}=8^{\frac{1}{3}}\).

Step2: Apply power - of - a - power rule

The power - of - a - power rule states that \((a^{m})^{n}=a^{mn}\). Here, \(a = 8\), \(m=\frac{1}{3}\), and \(n = 4\). Then \((8^{\frac{1}{3}})^{4}=8^{\frac{1}{3}\times4}=8^{\frac{4}{3}}\).

Answer:

\(8^{\frac{4}{3}}\) (assuming the last option "8^(4/3)" is what is meant by "8^(4/3)" in the original list of options)