Algebra Examples

$\frac{\sqrt{w^{11}}}{\sqrt[3]{w^{8}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $w^{11}$ as ${(w^{5})}^{2} w$ .

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Step 1.1.1

Factor out $w^{10}$ .

$\frac{\sqrt{w^{10} w}}{\sqrt[3]{w^{8}}}$

Step 1.1.2

Rewrite $w^{10}$ as ${(w^{5})}^{2}$ .

$\frac{\sqrt{{(w^{5})}^{2} w}}{\sqrt[3]{w^{8}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{w^{5} \sqrt{w}}{\sqrt[3]{w^{8}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $w^{8}$ as ${(w^{2})}^{3} w^{2}$ .

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Step 2.1.1

Factor out $w^{6}$ .

$\frac{w^{5} \sqrt{w}}{\sqrt[3]{w^{6} w^{2}}}$

Step 2.1.2

Rewrite $w^{6}$ as ${(w^{2})}^{3}$ .

$\frac{w^{5} \sqrt{w}}{\sqrt[3]{{(w^{2})}^{3} w^{2}}}$

Step 2.2

Pull terms out from under the radical.

$\frac{w^{5} \sqrt{w}}{w^{2} \sqrt[3]{w^{2}}}$

Step 3

Cancel the common factor of $w^{5}$ and $w^{2}$ .

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Step 3.1

Factor $w^{2}$ out of $w^{5} \sqrt{w}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $w^{2}$ out of $w^{2} \sqrt[3]{w^{2}}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

Raise $\sqrt[3]{w^{2}}$ to the power of $1$ .

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

Step 5.3

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

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

Step 5.4

Add $1$ and $2$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

Combine $\frac{2}{3}$ and $3$ .

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

Step 5.5.4

Multiply $2$ by $3$ .

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

Step 5.5.5

Cancel the common factor of $6$ and $3$ .

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Step 5.5.5.1

Factor $3$ out of $6$ .

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

Step 5.5.5.2

Cancel the common factors.

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Step 5.5.5.2.1

Factor $3$ out of $3$ .

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

Step 5.5.5.2.2

Cancel the common factor.

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

Step 5.5.5.2.3

Rewrite the expression.

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

Step 5.5.5.2.4

Divide $2$ by $1$ .

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

Step 6

Cancel the common factor of $w^{3}$ and $w^{2}$ .

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Step 6.1

Factor $w^{2}$ out of $w^{3} \sqrt{w} {\sqrt[3]{w^{2}}}^{2}$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Multiply by $1$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 6.2.4

Divide $w \sqrt{w} {\sqrt[3]{w^{2}}}^{2}$ by $1$ .

$w \sqrt{w} {\sqrt[3]{w^{2}}}^{2}$

Step 7

Rewrite ${\sqrt[3]{w^{2}}}^{2}$ as $\sqrt[3]{{(w^{2})}^{2}}$ .

$w \sqrt{w} \sqrt[3]{{(w^{2})}^{2}}$

Step 8

Multiply the exponents in ${(w^{2})}^{2}$ .

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Step 8.1

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

$w \sqrt{w} \sqrt[3]{w^{2 \cdot 2}}$

Step 8.2

Multiply $2$ by $2$ .

$w \sqrt{w} \sqrt[3]{w^{4}}$

Step 9

Factor out $w^{3}$ .

$w \sqrt{w} \sqrt[3]{w^{3} w}$

Step 10

Pull terms out from under the radical.

$w \sqrt{w} (w \sqrt[3]{w})$

Step 11

Multiply $w$ by $w$ by adding the exponents.

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Step 11.1

Move $w$ .

$w \cdot w \sqrt{w} \sqrt[3]{w}$

Step 11.2

Multiply $w$ by $w$ .

$w^{2} \sqrt{w} \sqrt[3]{w}$

Step 12

Multiply $w^{2} \sqrt{w} \sqrt[3]{w}$ .

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Step 12.1

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

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Step 12.1.1

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

$w^{2} (w^{\frac{1}{3}} \sqrt{w})$

Step 12.1.2

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

$w^{2} (w^{\frac{2}{6}} \sqrt{w})$

Step 12.1.3

Rewrite $w^{\frac{2}{6}}$ as $\sqrt[6]{w^{2}}$ .

$w^{2} (\sqrt[6]{w^{2}} \sqrt{w})$

Step 12.1.4

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

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

Step 12.1.5

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

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

Step 12.1.6

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

$w^{2} (\sqrt[6]{w^{2}} \sqrt[6]{w^{3}})$

Step 12.2

Combine using the product rule for radicals.

$w^{2} \sqrt[6]{w^{2} w^{3}}$

Step 12.3

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

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Step 12.3.1

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

$w^{2} \sqrt[6]{w^{2 + 3}}$

Step 12.3.2

Add $2$ and $3$ .

$w^{2} \sqrt[6]{w^{5}}$