Algebra Examples

$\frac{\sqrt[5]{h^{11}}}{\sqrt[6]{h^{5}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor out $h^{10}$ .

$\frac{\sqrt[5]{h^{10} h}}{\sqrt[6]{h^{5}}}$

Step 1.1.2

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

$\frac{\sqrt[5]{{(h^{2})}^{5} h}}{\sqrt[6]{h^{5}}}$

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{h^{2} \sqrt[5]{h}}{\sqrt[6]{h^{5}}}$ by $\frac{{\sqrt[6]{h^{5}}}^{5}}{{\sqrt[6]{h^{5}}}^{5}}$ .

$\frac{h^{2} \sqrt[5]{h}}{\sqrt[6]{h^{5}}} \cdot \frac{{\sqrt[6]{h^{5}}}^{5}}{{\sqrt[6]{h^{5}}}^{5}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{h^{2} \sqrt[5]{h}}{\sqrt[6]{h^{5}}}$ by $\frac{{\sqrt[6]{h^{5}}}^{5}}{{\sqrt[6]{h^{5}}}^{5}}$ .

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

Step 3.2

Raise $\sqrt[6]{h^{5}}$ to the power of $1$ .

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{{\sqrt[6]{h^{5}}}^{1} {\sqrt[6]{h^{5}}}^{5}}$

Step 3.3

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

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{{\sqrt[6]{h^{5}}}^{1 + 5}}$

Step 3.4

Add $1$ and $5$ .

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

Step 3.5

Rewrite ${\sqrt[6]{h^{5}}}^{6}$ as $h^{5}$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{h^{5}}$ as $h^{\frac{5}{6}}$ .

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{{(h^{\frac{5}{6}})}^{6}}$

Step 3.5.2

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

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{h^{\frac{5}{6} \cdot 6}}$

Step 3.5.3

Combine $\frac{5}{6}$ and $6$ .

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{h^{\frac{5 \cdot 6}{6}}}$

Step 3.5.4

Multiply $5$ by $6$ .

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{h^{\frac{30}{6}}}$

Step 3.5.5

Cancel the common factor of $30$ and $6$ .

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Step 3.5.5.1

Factor $6$ out of $30$ .

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{h^{\frac{6 \cdot 5}{6}}}$

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Factor $6$ out of $6$ .

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{h^{\frac{6 \cdot 5}{6 (1)}}}$

Step 3.5.5.2.2

Cancel the common factor.

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{h^{\frac{6 \cdot 5}{6 \cdot 1}}}$

Step 3.5.5.2.3

Rewrite the expression.

$\frac{h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}}{h^{\frac{5}{1}}}$

Step 3.5.5.2.4

Divide $5$ by $1$ .

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

Step 4

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

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Step 4.1

Factor $h^{2}$ out of $h^{2} \sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5}$ .

$\frac{h^{2} (\sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5})}{h^{5}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $h^{2}$ out of $h^{5}$ .

$\frac{h^{2} (\sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5})}{h^{2} h^{3}}$

Step 4.2.2

Cancel the common factor.

$\frac{h^{2} (\sqrt[5]{h} {\sqrt[6]{h^{5}}}^{5})}{h^{2} h^{3}}$

Step 4.2.3

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[6]{h^{5}}}^{5}$ as $\sqrt[6]{{(h^{5})}^{5}}$ .

$\frac{\sqrt[5]{h} \sqrt[6]{{(h^{5})}^{5}}}{h^{3}}$

Step 5.2

Multiply the exponents in ${(h^{5})}^{5}$ .

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Step 5.2.1

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

$\frac{\sqrt[5]{h} \sqrt[6]{h^{5 \cdot 5}}}{h^{3}}$

Step 5.2.2

Multiply $5$ by $5$ .

$\frac{\sqrt[5]{h} \sqrt[6]{h^{25}}}{h^{3}}$

Step 5.3

Rewrite $h^{25}$ as ${(h^{4})}^{6} h$ .

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Step 5.3.1

Factor out $h^{24}$ .

$\frac{\sqrt[5]{h} \sqrt[6]{h^{24} h}}{h^{3}}$

Step 5.3.2

Rewrite $h^{24}$ as ${(h^{4})}^{6}$ .

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

Step 5.4

Pull terms out from under the radical.

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

Step 5.5

Remove non-negative terms from the absolute value.

$\frac{\sqrt[5]{h} h^{4} \sqrt[6]{h}}{h^{3}}$

Step 5.6

Combine exponents.

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Step 5.6.1

Rewrite the expression using the least common index of $30$ .

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Step 5.6.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{h}$ as $h^{\frac{1}{5}}$ .

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

Step 5.6.1.2

Rewrite $h^{\frac{1}{5}}$ as $h^{\frac{6}{30}}$ .

$\frac{h^{4} (h^{\frac{6}{30}} \sqrt[6]{h})}{h^{3}}$

Step 5.6.1.3

Rewrite $h^{\frac{6}{30}}$ as $\sqrt[30]{h^{6}}$ .

$\frac{h^{4} (\sqrt[30]{h^{6}} \sqrt[6]{h})}{h^{3}}$

Step 5.6.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{h}$ as $h^{\frac{1}{6}}$ .

$\frac{h^{4} (\sqrt[30]{h^{6}} h^{\frac{1}{6}})}{h^{3}}$

Step 5.6.1.5

Rewrite $h^{\frac{1}{6}}$ as $h^{\frac{5}{30}}$ .

$\frac{h^{4} (\sqrt[30]{h^{6}} h^{\frac{5}{30}})}{h^{3}}$

Step 5.6.1.6

Rewrite $h^{\frac{5}{30}}$ as $\sqrt[30]{h^{5}}$ .

$\frac{h^{4} (\sqrt[30]{h^{6}} \sqrt[30]{h^{5}})}{h^{3}}$

Step 5.6.2

Combine using the product rule for radicals.

$\frac{h^{4} \sqrt[30]{h^{6} h^{5}}}{h^{3}}$

Step 5.6.3

Multiply $h^{6}$ by $h^{5}$ by adding the exponents.

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Step 5.6.3.1

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

$\frac{h^{4} \sqrt[30]{h^{6 + 5}}}{h^{3}}$

Step 5.6.3.2

Add $6$ and $5$ .

$\frac{h^{4} \sqrt[30]{h^{11}}}{h^{3}}$

Step 6

Cancel the common factor of $h^{4}$ and $h^{3}$ .

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Step 6.1

Factor $h^{3}$ out of $h^{4} \sqrt[30]{h^{11}}$ .

$\frac{h^{3} (h \sqrt[30]{h^{11}})}{h^{3}}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Multiply by $1$ .

$\frac{h^{3} (h \sqrt[30]{h^{11}})}{h^{3} \cdot 1}$

Step 6.2.2

Cancel the common factor.

$\frac{h^{3} (h \sqrt[30]{h^{11}})}{h^{3} \cdot 1}$

Step 6.2.3

Rewrite the expression.

$\frac{h \sqrt[30]{h^{11}}}{1}$

Step 6.2.4

Divide $h \sqrt[30]{h^{11}}$ by $1$ .

$h \sqrt[30]{h^{11}}$