Question

c. write each expression using a rational exponent

1. $sqrt3{6}$
2. $sqrt4{8}$
3. $sqrt3{12}$
4. $sqrt4{15}$
5. $sqrt4{9}$
6. $sqrt5{13}$

Explanation:

Problem 1: $\sqrt[3]{6}$

Step 1: Recall the formula for converting radicals to rational exponents.

The formula is $\sqrt[n]{a}=a^{\frac{1}{n}}$, where $n$ is the index of the radical and $a$ is the radicand.
For $\sqrt[3]{6}$, $n = 3$ and $a=6$.

Step 2: Apply the formula.

Substitute $n = 3$ and $a = 6$ into the formula $\sqrt[n]{a}=a^{\frac{1}{n}}$.
We get $\sqrt[3]{6}=6^{\frac{1}{3}}$.

Problem 2: $\sqrt[4]{8}$

Step 1: Recall the radical - rational exponent formula.

The formula $\sqrt[n]{a}=a^{\frac{1}{n}}$ holds, where $n$ is the root (index of the radical) and $a$ is the number inside the radical (radicand).
For $\sqrt[4]{8}$, $n = 4$ and $a = 8$.

Step 2: Apply the formula.

Using $\sqrt[n]{a}=a^{\frac{1}{n}}$, substitute $n = 4$ and $a=8$.
We obtain $\sqrt[4]{8}=8^{\frac{1}{4}}$.

Problem 3: $\sqrt[3]{12}$

Step 1: Recall the conversion formula.

The formula for converting a radical $\sqrt[n]{a}$ to a rational exponent is $a^{\frac{1}{n}}$, where $n$ is the index of the radical and $a$ is the radicand.
For $\sqrt[3]{12}$, $n = 3$ and $a = 12$.

Step 2: Apply the formula.

Substitute $n=3$ and $a = 12$ into $\sqrt[n]{a}=a^{\frac{1}{n}}$.
We get $\sqrt[3]{12}=12^{\frac{1}{3}}$.

Problem 4: $\sqrt[4]{h^{3}}$

Answer:

s:

1. $6^{\frac{1}{3}}$
2. $8^{\frac{1}{4}}$
3. $12^{\frac{1}{3}}$
4. $h^{\frac{3}{4}}$
5. $9^{\frac{1}{4}}$
6. $m^{\frac{3}{5}}$