Write the Fraction in Simplest Form ( fourth root of 9 square root of 9)/( fourth root of 9^5)

$\frac{\sqrt[4]{9} \sqrt{9}}{\sqrt[4]{9^{5}}}$

Step 1

Combine $\sqrt[4]{9}$ and $\sqrt[4]{9^{5}}$ into a single radical.

$\sqrt[4]{\frac{9}{9^{5}}} \sqrt{9}$

Step 2

Reduce the expression $\frac{9}{9^{5}}$ by cancelling the common factors.

Factor $9$ out of $9$ .

$\sqrt[4]{\frac{9 (1)}{9^{5}}} \sqrt{9}$

Factor $9$ out of $9^{5}$ .

$\sqrt[4]{\frac{9 (1)}{9 \cdot 9^{4}}} \sqrt{9}$

Cancel the common factor.

$\sqrt[4]{\frac{9 \cdot 1}{9 \cdot 9^{4}}} \sqrt{9}$

Rewrite the expression.

$\sqrt[4]{\frac{1}{9^{4}}} \sqrt{9}$

Step 3

Raise $9$ to the power of $4$ .

$\sqrt[4]{\frac{1}{6561}} \sqrt{9}$

Step 4

Rewrite $\sqrt[4]{\frac{1}{6561}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{6561}}$ .

$\frac{\sqrt[4]{1}}{\sqrt[4]{6561}} \sqrt{9}$

Step 5

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[4]{6561}} \sqrt{9}$

Step 6

Simplify the denominator.

Rewrite $6561$ as $9^{4}$ .

$\frac{1}{\sqrt[4]{9^{4}}} \sqrt{9}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{9} \sqrt{9}$

Step 7

Rewrite $9$ as $3^{2}$ .

$\frac{1}{9} \sqrt{3^{2}}$

Step 8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{9} \cdot 3$

Step 9

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$\frac{1}{3 (3)} \cdot 3$

Cancel the common factor.

$\frac{1}{3 \cdot 3} \cdot 3$

Rewrite the expression.

$\frac{1}{3}$