Question

use rational exponents to simplify the radical expression. write final answers in radical form when necessary.
$sqrt6{8}$
$sqrt6{8} = \square$ (type an exact answer, using radicals as needed.)

Explanation:

Step1: Rewrite radical as exponent

Recall that \(\sqrt[n]{a}=a^{\frac{1}{n}}\), so \(\sqrt[6]{8}=8^{\frac{1}{6}}\).

Step2: Express 8 as a power of 2

Since \(8 = 2^3\), substitute this into the expression: \((2^3)^{\frac{1}{6}}\).

Step3: Apply exponent rule \((a^m)^n=a^{mn}\)

Multiply the exponents: \(2^{3\times\frac{1}{6}} = 2^{\frac{1}{2}}\).

Step4: Rewrite back to radical form

Since \(a^{\frac{1}{n}}=\sqrt[n]{a}\), \(2^{\frac{1}{2}}=\sqrt{2}\).

Answer:

\(\sqrt{2}\)