Simplify ( fourth root of y^3)/( sixth root of y^3)

$\frac{\sqrt[4]{y^{3}}}{\sqrt[6]{y^{3}}}$

Step 1

Simplify the denominator.

Rewrite $\sqrt[6]{y^{3}}$ as $\sqrt{\sqrt[3]{y^{3}}}$ .

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

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

$\frac{\sqrt[4]{y^{3}} \sqrt{y}}{{\sqrt{y}}^{2}}$

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{y^{3}} \sqrt{y}}{{(y^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt[4]{y^{3}} \sqrt{y}}{y^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt[4]{y^{3}} \sqrt{y}}{y^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt[4]{y^{3}} \sqrt{y}}{y^{\frac{2}{2}}}$

Rewrite the expression.

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

Simplify.

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

Step 4

Simplify the numerator.

Rewrite the expression using the least common index of $4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{y^{3}} y^{\frac{1}{2}}}{y}$

Rewrite $y^{\frac{1}{2}}$ as $y^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{y^{3}} y^{\frac{2}{4}}}{y}$

Rewrite $y^{\frac{2}{4}}$ as $\sqrt[4]{y^{2}}$ .

$\frac{\sqrt[4]{y^{3}} \sqrt[4]{y^{2}}}{y}$

Combine using the product rule for radicals.

$\frac{\sqrt[4]{y^{3} y^{2}}}{y}$

Multiply $y^{3}$ by $y^{2}$ by adding the exponents.

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

$\frac{\sqrt[4]{y^{3 + 2}}}{y}$

Add $3$ and $2$ .

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

Step 5

Simplify the numerator.

Factor out $y^{4}$ .

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

Pull terms out from under the radical.

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

Step 6

Cancel the common factor of $y$ .

Cancel the common factor.

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

Divide $\sqrt[4]{y}$ by $1$ .

$\sqrt[4]{y}$

Simplify ( fourth root of y^3)/( sixth root of y^3)

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