Algebra Examples

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

Step 1

Multiply $\frac{\sqrt[6]{j^{5}}}{\sqrt[5]{j^{4}}}$ by $\frac{{\sqrt[5]{j^{4}}}^{4}}{{\sqrt[5]{j^{4}}}^{4}}$ .

$\frac{\sqrt[6]{j^{5}}}{\sqrt[5]{j^{4}}} \cdot \frac{{\sqrt[5]{j^{4}}}^{4}}{{\sqrt[5]{j^{4}}}^{4}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[6]{j^{5}}}{\sqrt[5]{j^{4}}}$ by $\frac{{\sqrt[5]{j^{4}}}^{4}}{{\sqrt[5]{j^{4}}}^{4}}$ .

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

Step 2.2

Raise $\sqrt[5]{j^{4}}$ to the power of $1$ .

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{{\sqrt[5]{j^{4}}}^{1} {\sqrt[5]{j^{4}}}^{4}}$

Step 2.3

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

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{{\sqrt[5]{j^{4}}}^{1 + 4}}$

Step 2.4

Add $1$ and $4$ .

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

Step 2.5

Rewrite ${\sqrt[5]{j^{4}}}^{5}$ as $j^{4}$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{j^{4}}$ as $j^{\frac{4}{5}}$ .

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

Step 2.5.2

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

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{j^{\frac{4}{5} \cdot 5}}$

Step 2.5.3

Combine $\frac{4}{5}$ and $5$ .

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{j^{\frac{4 \cdot 5}{5}}}$

Step 2.5.4

Multiply $4$ by $5$ .

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{j^{\frac{20}{5}}}$

Step 2.5.5

Cancel the common factor of $20$ and $5$ .

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Step 2.5.5.1

Factor $5$ out of $20$ .

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{j^{\frac{5 \cdot 4}{5}}}$

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $5$ out of $5$ .

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{j^{\frac{5 \cdot 4}{5 (1)}}}$

Step 2.5.5.2.2

Cancel the common factor.

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{j^{\frac{5 \cdot 4}{5 \cdot 1}}}$

Step 2.5.5.2.3

Rewrite the expression.

$\frac{\sqrt[6]{j^{5}} {\sqrt[5]{j^{4}}}^{4}}{j^{\frac{4}{1}}}$

Step 2.5.5.2.4

Divide $4$ by $1$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[5]{j^{4}}}^{4}$ as $\sqrt[5]{{(j^{4})}^{4}}$ .

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

Step 3.2

Multiply the exponents in ${(j^{4})}^{4}$ .

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Step 3.2.1

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

$\frac{\sqrt[6]{j^{5}} \sqrt[5]{j^{4 \cdot 4}}}{j^{4}}$

Step 3.2.2

Multiply $4$ by $4$ .

$\frac{\sqrt[6]{j^{5}} \sqrt[5]{j^{16}}}{j^{4}}$

Step 3.3

Rewrite $j^{16}$ as ${(j^{3})}^{5} j$ .

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Step 3.3.1

Factor out $j^{15}$ .

$\frac{\sqrt[6]{j^{5}} \sqrt[5]{j^{15} j}}{j^{4}}$

Step 3.3.2

Rewrite $j^{15}$ as ${(j^{3})}^{5}$ .

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

Step 3.4

Pull terms out from under the radical.

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

Step 3.5

Combine exponents.

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Step 3.5.1

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

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Step 3.5.1.1

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

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

Step 3.5.1.2

Rewrite $j^{\frac{5}{6}}$ as $j^{\frac{25}{30}}$ .

$\frac{j^{3} (j^{\frac{25}{30}} \sqrt[5]{j})}{j^{4}}$

Step 3.5.1.3

Rewrite $j^{\frac{25}{30}}$ as $\sqrt[30]{j^{25}}$ .

$\frac{j^{3} (\sqrt[30]{j^{25}} \sqrt[5]{j})}{j^{4}}$

Step 3.5.1.4

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

$\frac{j^{3} (\sqrt[30]{j^{25}} j^{\frac{1}{5}})}{j^{4}}$

Step 3.5.1.5

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

$\frac{j^{3} (\sqrt[30]{j^{25}} j^{\frac{6}{30}})}{j^{4}}$

Step 3.5.1.6

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

$\frac{j^{3} (\sqrt[30]{j^{25}} \sqrt[30]{j^{6}})}{j^{4}}$

Step 3.5.2

Combine using the product rule for radicals.

$\frac{j^{3} \sqrt[30]{j^{25} j^{6}}}{j^{4}}$

Step 3.5.3

Multiply $j^{25}$ by $j^{6}$ by adding the exponents.

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Step 3.5.3.1

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

$\frac{j^{3} \sqrt[30]{j^{25 + 6}}}{j^{4}}$

Step 3.5.3.2

Add $25$ and $6$ .

$\frac{j^{3} \sqrt[30]{j^{31}}}{j^{4}}$

Step 3.6

Factor out $j^{30}$ .

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

Step 3.7

Pull terms out from under the radical.

$\frac{j^{3} (| j | \sqrt[30]{j})}{j^{4}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite.

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

Step 4.2

Rewrite ${\sqrt[5]{j^{4}}}^{4}$ as $\sqrt[5]{{(j^{4})}^{4}}$ .

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

Step 4.3

Multiply the exponents in ${(j^{4})}^{4}$ .

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Step 4.3.1

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

$\frac{\sqrt[6]{j^{5}} \sqrt[5]{j^{4 \cdot 4}}}{j^{4}}$

Step 4.3.2

Multiply $4$ by $4$ .

$\frac{\sqrt[6]{j^{5}} \sqrt[5]{j^{16}}}{j^{4}}$

Step 4.4

Rewrite $j^{16}$ as ${(j^{3})}^{5} j$ .

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Step 4.4.1

Factor out $j^{15}$ .

$\frac{\sqrt[6]{j^{5}} \sqrt[5]{j^{15} j}}{j^{4}}$

Step 4.4.2

Rewrite $j^{15}$ as ${(j^{3})}^{5}$ .

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

Step 4.5

Pull terms out from under the radical.

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

Step 4.6

Combine exponents.

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Step 4.6.1

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

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Step 4.6.1.1

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

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

Step 4.6.1.2

Rewrite $j^{\frac{5}{6}}$ as $j^{\frac{25}{30}}$ .

$\frac{j^{3} (j^{\frac{25}{30}} \sqrt[5]{j})}{j^{4}}$

Step 4.6.1.3

Rewrite $j^{\frac{25}{30}}$ as $\sqrt[30]{j^{25}}$ .

$\frac{j^{3} (\sqrt[30]{j^{25}} \sqrt[5]{j})}{j^{4}}$

Step 4.6.1.4

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

$\frac{j^{3} (\sqrt[30]{j^{25}} j^{\frac{1}{5}})}{j^{4}}$

Step 4.6.1.5

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

$\frac{j^{3} (\sqrt[30]{j^{25}} j^{\frac{6}{30}})}{j^{4}}$

Step 4.6.1.6

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

$\frac{j^{3} (\sqrt[30]{j^{25}} \sqrt[30]{j^{6}})}{j^{4}}$

Step 4.6.2

Combine using the product rule for radicals.

$\frac{j^{3} \sqrt[30]{j^{25} j^{6}}}{j^{4}}$

Step 4.6.3

Multiply $j^{25}$ by $j^{6}$ by adding the exponents.

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Step 4.6.3.1

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

$\frac{j^{3} \sqrt[30]{j^{25 + 6}}}{j^{4}}$

Step 4.6.3.2

Add $25$ and $6$ .

$\frac{j^{3} \sqrt[30]{j^{31}}}{j^{4}}$

Step 4.7

Factor out $j^{30}$ .

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

Step 4.8

Pull terms out from under the radical.

$\frac{j^{3} (| j | \sqrt[30]{j})}{j^{4}}$

Step 4.9

Remove unnecessary parentheses.

$\frac{j^{3} | j | \sqrt[30]{j}}{j^{4}}$

Step 5

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

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Step 5.1

Factor $j^{3}$ out of $j^{3} | j | \sqrt[30]{j}$ .

$\frac{j^{3} (| j | \sqrt[30]{j})}{j^{4}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $j^{3}$ out of $j^{4}$ .

$\frac{j^{3} (| j | \sqrt[30]{j})}{j^{3} j}$

Step 5.2.2

Cancel the common factor.

$\frac{j^{3} (| j | \sqrt[30]{j})}{j^{3} j}$

Step 5.2.3

Rewrite the expression.

$\frac{| j | \sqrt[30]{j}}{j}$