Question: Rewrite the following radical with a rational exponent. \[ \left( \sqrt[7]{8x} \right)^4 \]…

Rewrite the following radical with a rational exponent.

\[ \left( \sqrt[7]{8x} \right)^4 \]

\[ \left( \sqrt[7]{8x} \right)^4 = \boxed{\phantom{xxxx}} \]

(Simplify your answer. Use integers or fractions for any numbers in the expression.)

Solution

To solve the problem, we need to rewrite the expression \(\left( 7 \sqrt[4]{8x} \right)^4\) with a rational exponent. First, express the root as a rational exponent: \[ \sqrt[4]{8x} = (8x)^{1/4} \] Now substitute this into the original expression: \[ \left( 7 (8x)^{1/4} \right)^4 \] Apply the exponent to both the 7 and the \((8x)^{1/4}\): \[ 7^4 \times \left((8x)^{1/4}\right)^4 \] Simplify by multiplying the exponents: \[ 7^4 \times (8x)^{(1/4) \times 4} \] This simplifies to: \[ 7^4 \times (8x)^1 \] Now calculate \(7^4\): \[ 7^4 = 2401 \] So the simplified expression is: \[ 2401 \times 8x = 19208x \] The final answer is: \[ 19208x \]