Question

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

answer attempt 4 out of 4

Explanation:

Step1: Recall the exponent rule

The rule for converting a fractional exponent to a radical is \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\). For \(a^{\frac{1}{n}}\), this simplifies to \(\sqrt[n]{a}\) since \(m = 1\).
So for \(8^{\frac{1}{5}}\), using the rule \(a^{\frac{1}{n}}=\sqrt[n]{a}\) where \(a = 8\) and \(n=5\).

Step2: Write the radical form

Substituting the values, we get \(\sqrt[5]{8}\). Since \(8\) cannot be simplified further under a fifth - root (because \(8 = 2^3\) and \(3<5\), and there are no perfect fifth - power factors of \(8\)), the simplest radical form of \(8^{\frac{1}{5}}\) is \(\sqrt[5]{8}\).

Answer:

\(\sqrt[5]{8}\)