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

Multiply $\frac{| w^{5} | \sqrt{w}}{w^{2} \sqrt[3]{w^{2}}}$ by $\frac{{\sqrt[3]{w^{2}}}^{2}}{{\sqrt[3]{w^{2}}}^{2}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{| w^{5} | \sqrt{w}}{w^{2} \sqrt[3]{w^{2}}}$ by $\frac{{\sqrt[3]{w^{2}}}^{2}}{{\sqrt[3]{w^{2}}}^{2}}$ .

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

Step 4.2

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

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

Step 4.3

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

$\frac{| w^{5} | \sqrt{w} {\sqrt[3]{w^{2}}}^{2}}{w^{2} ({\sqrt[3]{w^{2}}}^{1} {\sqrt[3]{w^{2}}}^{2})}$

Step 4.4

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

$\frac{| w^{5} | \sqrt{w} {\sqrt[3]{w^{2}}}^{2}}{w^{2} {\sqrt[3]{w^{2}}}^{1 + 2}}$

Step 4.5

Add $1$ and $2$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Multiply $2$ by $3$ .

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

Step 4.6.5

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

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Step 4.6.5.1

Factor $3$ out of $6$ .

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

Step 4.6.5.2

Cancel the common factors.

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Step 4.6.5.2.1

Factor $3$ out of $3$ .

$\frac{| w^{5} | \sqrt{w} {\sqrt[3]{w^{2}}}^{2}}{w^{2} w^{\frac{3 \cdot 2}{3 (1)}}}$

Step 4.6.5.2.2

Cancel the common factor.

$\frac{| w^{5} | \sqrt{w} {\sqrt[3]{w^{2}}}^{2}}{w^{2} w^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Step 4.6.5.2.3

Rewrite the expression.

$\frac{| w^{5} | \sqrt{w} {\sqrt[3]{w^{2}}}^{2}}{w^{2} w^{\frac{2}{1}}}$

Step 4.6.5.2.4

Divide $2$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Add $2$ and $2$ .

$\frac{| w^{5} | \sqrt{w} {\sqrt[3]{w^{2}}}^{2}}{w^{4}}$

Step 6

Simplify the numerator.

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Step 6.1

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

$\frac{| w^{5} | \sqrt{w} \sqrt[3]{{(w^{2})}^{2}}}{w^{4}}$

Step 6.2

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

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Step 6.2.1

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

$\frac{| w^{5} | \sqrt{w} \sqrt[3]{w^{2 \cdot 2}}}{w^{4}}$

Step 6.2.2

Multiply $2$ by $2$ .

$\frac{| w^{5} | \sqrt{w} \sqrt[3]{w^{4}}}{w^{4}}$

Step 6.3

Factor out $w^{3}$ .

$\frac{| w^{5} | \sqrt{w} \sqrt[3]{w^{3} w}}{w^{4}}$

Step 6.4

Pull terms out from under the radical.

$\frac{| w^{5} | \sqrt{w} w \sqrt[3]{w}}{w^{4}}$

Step 6.5

Combine exponents.

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Step 6.5.1

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

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Step 6.5.1.1

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

$\frac{| w^{5} | w (w^{\frac{1}{2}} \sqrt[3]{w})}{w^{4}}$

Step 6.5.1.2

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

$\frac{| w^{5} | w (w^{\frac{3}{6}} \sqrt[3]{w})}{w^{4}}$

Step 6.5.1.3

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

$\frac{| w^{5} | w (\sqrt[6]{w^{3}} \sqrt[3]{w})}{w^{4}}$

Step 6.5.1.4

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

$\frac{| w^{5} | w (\sqrt[6]{w^{3}} w^{\frac{1}{3}})}{w^{4}}$

Step 6.5.1.5

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

$\frac{| w^{5} | w (\sqrt[6]{w^{3}} w^{\frac{2}{6}})}{w^{4}}$

Step 6.5.1.6

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

$\frac{| w^{5} | w (\sqrt[6]{w^{3}} \sqrt[6]{w^{2}})}{w^{4}}$

Step 6.5.2

Combine using the product rule for radicals.

$\frac{| w^{5} | w \sqrt[6]{w^{3} w^{2}}}{w^{4}}$

Step 6.5.3

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

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Step 6.5.3.1

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

$\frac{| w^{5} | w \sqrt[6]{w^{3 + 2}}}{w^{4}}$

Step 6.5.3.2

Add $3$ and $2$ .

$\frac{| w^{5} | w \sqrt[6]{w^{5}}}{w^{4}}$

Step 7

Cancel the common factor of $w$ and $w^{4}$ .

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Step 7.1

Factor $w$ out of $| w^{5} | w \sqrt[6]{w^{5}}$ .

$\frac{w (| w^{5} | \sqrt[6]{w^{5}})}{w^{4}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $w$ out of $w^{4}$ .

$\frac{w (| w^{5} | \sqrt[6]{w^{5}})}{w \cdot w^{3}}$

Step 7.2.2

Cancel the common factor.

$\frac{w (| w^{5} | \sqrt[6]{w^{5}})}{w \cdot w^{3}}$

Step 7.2.3

Rewrite the expression.

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