Question: Rewrite the following radical with a rational exponent. \[ \sqrt{\left(\frac{3x}{y}\right)^5}…

Rewrite the following radical with a rational exponent.

\[ \sqrt{\left(\frac{3x}{y}\right)^5} \]

\[ \sqrt{\left(\frac{3x}{y}\right)^5} = \boxed{\phantom{xxxx}} \]

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

Solution

To rewrite the given radical expression with a rational exponent, start by recognizing the components of the radical. The expression is: \[ \sqrt{ \left( \frac{3x}{y} \right)^5 } \] Step 1: Rewrite the square root as a rational exponent. The square root of a number is the same as raising that number to the power of \( \frac{1}{2} \). Thus, the expression becomes: \[ \left( \frac{3x}{y} \right)^{5 \cdot \frac{1}{2}} \] Step 2: Multiply the exponents. Multiply the exponents \(5\) and \( \frac{1}{2} \): \[ 5 \cdot \frac{1}{2} = \frac{5}{2} \] Step 3: Write the final expression. The expression with a rational exponent is: \[ \left( \frac{3x}{y} \right)^{\frac{5}{2}} \]