Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

$\frac{\sqrt[6]{y} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1} {\sqrt[3]{y}}^{2}}$

Step 2.3

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

$\frac{\sqrt[6]{y} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1 + 2}}$

Step 2.4

Add $1$ and $2$ .

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

Step 2.5

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

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Step 2.5.1

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

$\frac{\sqrt[6]{y} {\sqrt[3]{y}}^{2}}{{(y^{\frac{1}{3}})}^{3}}$

Step 2.5.2

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

$\frac{\sqrt[6]{y} {\sqrt[3]{y}}^{2}}{y^{\frac{1}{3} \cdot 3}}$

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[6]{y} {\sqrt[3]{y}}^{2}}{y^{1}}$

Step 2.5.5

Simplify.

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

Step 3

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

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

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

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Step 4.3.1

Multiply $y$ by $y^{4}$ .

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Step 4.3.1.1

Raise $y$ to the power of $1$ .

$\frac{\sqrt[6]{y^{1} y^{4}}}{y}$

Step 4.3.1.2

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

$\frac{\sqrt[6]{y^{1 + 4}}}{y}$

Step 4.3.2

Add $1$ and $4$ .

$\frac{\sqrt[6]{y^{5}}}{y}$