Question

Write with a rational exponent:

(a)$\sqrt{10m}$

(b)$\sqrt[5]{3n}$

(c) $3\sqrt[4]{6y}$

Short answer

The rational exponent of $\sqrt{10m}$is${\left(10m\right)}^{\frac{1}{2}}$

The rational exponent of $\sqrt[5]{3n}$is${\left(3n\right)}^{\frac{1}{5}}$

The rational exponent of $3\sqrt[4]{6y}$is $3{\left(6y\right)}^{\frac{1}{4}}$

Step-by-step solution

Part (a) Given

The radical expression $\sqrt{10m}$

To write each radical in the form of $a^{\frac{1}{n}}$

Part (a) Step 2. Write as rational exponent

No index is shown in $\sqrt{10m}$

The denominator of the exponent will be $2$.

Put parentheses around the entire expression.

The rational exponent is ${\left(10m\right)}^{\frac{1}{2}}$

Part (b) Step 1. Given

The radical expression is $\sqrt[5]{3n}$

To write each radical in the form of $a^{\frac{1}{n}}$

Part (b) Step 2. Write as rational exponent

The index is $4$, so the denominator of the exponent will be $5$.

Put parentheses around the entire expression.

The rational exponent is${\left(3n\right)}^{\frac{1}{5}}$

Part (c) Step 1. Given

The radical expression is$3\sqrt[4]{6y}$

To write each radical in the form of$a^{\frac{1}{n}}$

Part (c) Step 2. Write as rational exponent

The index is $4$, so the denominator of the exponent will be $4$.

Put parentheses around the entire expression.

The rational exponent is$3{\left(6y\right)}^{\frac{1}{4}}$