Simplify fourth root of 81/y

$\sqrt[4]{\frac{81}{y}}$

Step 1

Rewrite $\sqrt[4]{\frac{81}{y}}$ as $\frac{\sqrt[4]{81}}{\sqrt[4]{y}}$ .

$\frac{\sqrt[4]{81}}{\sqrt[4]{y}}$

Step 2

Simplify the numerator.

Rewrite $81$ as $3^{4}$ .

$\frac{\sqrt[4]{3^{4}}}{\sqrt[4]{y}}$

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

$\frac{3}{\sqrt[4]{y}}$

Step 3

Multiply $\frac{3}{\sqrt[4]{y}}$ by $\frac{{\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{3}}$ .

$\frac{3}{\sqrt[4]{y}} \cdot \frac{{\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{3}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{3}{\sqrt[4]{y}}$ by $\frac{{\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{3}}$ .

$\frac{3 {\sqrt[4]{y}}^{3}}{\sqrt[4]{y} {\sqrt[4]{y}}^{3}}$

Raise $\sqrt[4]{y}$ to the power of $1$ .

$\frac{3 {\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{1} {\sqrt[4]{y}}^{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 {\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{3 {\sqrt[4]{y}}^{3}}{{\sqrt[4]{y}}^{4}}$

Rewrite ${\sqrt[4]{y}}^{4}$ as $y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y}$ as $y^{\frac{1}{4}}$ .

$\frac{3 {\sqrt[4]{y}}^{3}}{{(y^{\frac{1}{4}})}^{4}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 {\sqrt[4]{y}}^{3}}{y^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$\frac{3 {\sqrt[4]{y}}^{3}}{y^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{3 {\sqrt[4]{y}}^{3}}{y^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{3 {\sqrt[4]{y}}^{3}}{y^{1}}$

Simplify.

$\frac{3 {\sqrt[4]{y}}^{3}}{y}$

Step 5

Rewrite ${\sqrt[4]{y}}^{3}$ as ${(y^{3})}^{\frac{1}{4}}$ .

$\frac{3 \sqrt[4]{y^{3}}}{y}$