Question

rewrite using exponents:

1. \\(\sqrt{5^7}\\)
2. \\(\sqrt4{x^9}\\)
3. \\(\sqrt13{w}\\)
4. \\(\sqrt3{8^{92}}\\)

Explanation:

Step1: Apply root-to-exponent rule

Recall $\sqrt[n]{a^m}=a^{\frac{m}{n}}$; for $\sqrt{5^7}$, $n=2$, $m=7$.
$\sqrt{5^7}=5^{\frac{7}{2}}$

Step2: Apply root-to-exponent rule

For $\sqrt[4]{x^9}$, $n=4$, $m=9$.
$\sqrt[4]{x^9}=x^{\frac{9}{4}}$

Step3: Apply root-to-exponent rule

For $\sqrt[13]{w}$, $w=w^1$, so $n=13$, $m=1$.
$\sqrt[13]{w}=w^{\frac{1}{13}}$

Step4: Apply root-to-exponent rule

For $\sqrt[3]{8^{92}}$, $n=3$, $m=92$.
$\sqrt[3]{8^{92}}=8^{\frac{92}{3}}$

Answer:

1. $5^{\frac{7}{2}}$
2. $x^{\frac{9}{4}}$
3. $w^{\frac{1}{13}}$
4. $8^{\frac{92}{3}}$