Question

practice
see example 1
write each radical using rational exponents.

1. $sqrt{3}$
2. $sqrt3{7}$
3. $sqrt3{3^2}$
4. $sqrt4{3^5}$

ponents to determine

Explanation:

Step1: Convert root to rational exponent

For $\sqrt{5}$, recall $\sqrt[n]{a}=a^{\frac{1}{n}}$. Here $n=2$, so:
$\sqrt{5}=5^{\frac{1}{2}}$

Step2: Convert root to rational exponent

For $\frac{1}{\sqrt{7}}$, rewrite as $\sqrt{7}^{-1}$, then apply $\sqrt[n]{a}=a^{\frac{1}{n}}$:
$\frac{1}{\sqrt{7}}=7^{-\frac{1}{2}}$

Step3: Convert root to rational exponent

For $\sqrt[4]{3^2}$, use $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. Here $n=4, m=2$:
$\sqrt[4]{3^2}=3^{\frac{2}{4}}=3^{\frac{1}{2}}$

Step4: Convert root to rational exponent

For $\sqrt[3]{x^5}$, use $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. Here $n=3, m=5$:
$\sqrt[3]{x^5}=x^{\frac{5}{3}}$

Answer:

1. $5^{\frac{1}{2}}$
2. $7^{-\frac{1}{2}}$
3. $3^{\frac{1}{2}}$
4. $x^{\frac{5}{3}}$