Question

select the expression equivalent to \\(sqrt9{x^{74}}\\). answer \\(x^{8}sqrt9{x^{2}}\\) \\(x^{8}sqrt9{x}\\) \\(x^{7}sqrt9{x^{2}}\\) \\(x^{7}sqrt9{x}\\)

Explanation:

Step1: Rewrite root as exponent

$\sqrt[9]{x^{74}} = x^{\frac{74}{9}}$

Step2: Split exponent into integer+fraction

$\frac{74}{9} = 8 + \frac{2}{9}$, so $x^{\frac{74}{9}} = x^{8+\frac{2}{9}}$

Step3: Separate exponents via product rule

$x^{8+\frac{2}{9}} = x^8 \cdot x^{\frac{2}{9}}$

Step4: Rewrite fractional exponent as root

$x^{\frac{2}{9}} = \sqrt[9]{x^2}$, so $x^8 \cdot x^{\frac{2}{9}} = x^8\sqrt[9]{x^2}$

Answer:

$x^8\sqrt[9]{x^2}$