Simplify ( fourth root of y^2)/( sixth root of y^2)

$\frac{\sqrt[4]{y^{2}}}{\sqrt[6]{y^{2}}}$

Step 1

Simplify the numerator.

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

$\frac{\sqrt{\sqrt{y^{2}}}}{\sqrt[6]{y^{2}}}$

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

$\frac{\sqrt{y}}{\sqrt[6]{y^{2}}}$

Step 2

Simplify the denominator.

Rewrite $\sqrt[6]{y^{2}}$ as $\sqrt[3]{\sqrt{y^{2}}}$ .

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

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

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

Step 3

Multiply $\frac{\sqrt{y}}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

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

Step 4

Combine and simplify the denominator.

Multiply $\frac{\sqrt{y}}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

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

Raise $\sqrt[3]{y}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

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

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

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

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 5

Rewrite ${\sqrt[3]{y}}^{2}$ as $\sqrt[3]{y^{2}}$ .

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

Step 6

Simplify the numerator.

Rewrite the expression using the least common index of $6$ .

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

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

Rewrite $y^{\frac{1}{2}}$ as $y^{\frac{3}{6}}$ .

$\frac{y^{\frac{3}{6}} \sqrt[3]{y^{2}}}{y}$

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

$\frac{\sqrt[6]{y^{3}} \sqrt[3]{y^{2}}}{y}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{2}}$ as $y^{\frac{2}{3}}$ .

$\frac{\sqrt[6]{y^{3}} y^{\frac{2}{3}}}{y}$

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{4}{6}}$ .

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

Rewrite $y^{\frac{4}{6}}$ as $\sqrt[6]{y^{4}}$ .

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

Combine using the product rule for radicals.

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

Multiply $y^{3}$ by $y^{4}$ by adding the exponents.

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

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

Add $3$ and $4$ .

$\frac{\sqrt[6]{y^{7}}}{y}$

Step 7

Simplify the numerator.

Factor out $y^{6}$ .

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

Pull terms out from under the radical.

$\frac{y \sqrt[6]{y}}{y}$

Step 8

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{y \sqrt[6]{y}}{y}$

Divide $\sqrt[6]{y}$ by $1$ .

$\sqrt[6]{y}$

Simplify ( fourth root of y^2)/( sixth root of y^2)

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