Algebra Examples

$\sqrt[5]{\frac{8 x^{3}}{y^{4}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 1

Rewrite $\sqrt[5]{\frac{8 x^{3}}{y^{4}}}$ as $\frac{\sqrt[5]{8 x^{3}}}{\sqrt[5]{y^{4}}}$ .

$\frac{\sqrt[5]{8 x^{3}}}{\sqrt[5]{y^{4}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 2

Multiply $\frac{\sqrt[5]{8 x^{3}}}{\sqrt[5]{y^{4}}}$ by $\frac{{\sqrt[5]{y^{4}}}^{4}}{{\sqrt[5]{y^{4}}}^{4}}$ .

$\frac{\sqrt[5]{8 x^{3}}}{\sqrt[5]{y^{4}}} \cdot \frac{{\sqrt[5]{y^{4}}}^{4}}{{\sqrt[5]{y^{4}}}^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[5]{8 x^{3}}}{\sqrt[5]{y^{4}}}$ by $\frac{{\sqrt[5]{y^{4}}}^{4}}{{\sqrt[5]{y^{4}}}^{4}}$ .

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{\sqrt[5]{y^{4}} {\sqrt[5]{y^{4}}}^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.2

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

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{{\sqrt[5]{y^{4}}}^{1} {\sqrt[5]{y^{4}}}^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.3

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

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{{\sqrt[5]{y^{4}}}^{1 + 4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.4

Add $1$ and $4$ .

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{{\sqrt[5]{y^{4}}}^{5}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5

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

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Step 3.5.1

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

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{{(y^{\frac{4}{5}})}^{5}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5.2

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

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{y^{\frac{4}{5} \cdot 5}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5.3

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

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{y^{\frac{4 \cdot 5}{5}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5.4

Multiply $4$ by $5$ .

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{y^{\frac{20}{5}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5.5

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

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Step 3.5.5.1

Factor $5$ out of $20$ .

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{y^{\frac{5 \cdot 4}{5}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5.5.2

Cancel the common factors.

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Step 3.5.5.2.1

Factor $5$ out of $5$ .

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{y^{\frac{5 \cdot 4}{5 (1)}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5.5.2.2

Cancel the common factor.

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{y^{\frac{5 \cdot 4}{5 \cdot 1}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5.5.2.3

Rewrite the expression.

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{y^{\frac{4}{1}}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 3.5.5.2.4

Divide $4$ by $1$ .

$\frac{\sqrt[5]{8 x^{3}} {\sqrt[5]{y^{4}}}^{4}}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{\sqrt[5]{8 x^{3}} \sqrt[5]{{(y^{4})}^{4}}}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{\sqrt[5]{8 x^{3}} \sqrt[5]{y^{4 \cdot 4}}}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 4.2.2

Multiply $4$ by $4$ .

$\frac{\sqrt[5]{8 x^{3}} \sqrt[5]{y^{16}}}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 4.3

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

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Step 4.3.1

Factor out $y^{15}$ .

$\frac{\sqrt[5]{8 x^{3}} \sqrt[5]{y^{15} y}}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 4.3.2

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

$\frac{\sqrt[5]{8 x^{3}} \sqrt[5]{{(y^{3})}^{5} y}}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 4.4

Pull terms out from under the radical.

$\frac{\sqrt[5]{8 x^{3}} y^{3} \sqrt[5]{y}}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 4.5

Combine using the product rule for radicals.

$\frac{y^{3} \sqrt[5]{8 x^{3} y}}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 5

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

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Step 5.1

Factor $y^{3}$ out of $y^{3} \sqrt[5]{8 x^{3} y}$ .

$\frac{y^{3} (\sqrt[5]{8 x^{3} y})}{y^{4}} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

$\frac{y^{3} (\sqrt[5]{8 x^{3} y})}{y^{3} y} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 5.2.2

Cancel the common factor.

$\frac{y^{3} \sqrt[5]{8 x^{3} y}}{y^{3} y} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 5.2.3

Rewrite the expression.

$\frac{\sqrt[5]{8 x^{3} y}}{y} \sqrt[5]{\frac{4 x^{4}}{y^{2}}}$

Step 6

Rewrite $\sqrt[5]{\frac{4 x^{4}}{y^{2}}}$ as $\frac{\sqrt[5]{4 x^{4}}}{\sqrt[5]{y^{2}}}$ .

$\frac{\sqrt[5]{8 x^{3} y}}{y} \cdot \frac{\sqrt[5]{4 x^{4}}}{\sqrt[5]{y^{2}}}$

Step 7

Combine.

$\frac{\sqrt[5]{8 x^{3} y} \sqrt[5]{4 x^{4}}}{y \sqrt[5]{y^{2}}}$

Step 8

Combine $\sqrt[5]{8 x^{3} y}$ and $\sqrt[5]{y^{2}}$ into a single radical.

$\frac{\sqrt[5]{\frac{8 x^{3} y}{y^{2}}} \sqrt[5]{4 x^{4}}}{y}$

Step 9

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

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Step 9.1

Factor $y$ out of $8 x^{3} y$ .

$\frac{\sqrt[5]{\frac{y (8 x^{3})}{y^{2}}} \sqrt[5]{4 x^{4}}}{y}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $y$ out of $y^{2}$ .

$\frac{\sqrt[5]{\frac{y (8 x^{3})}{y \cdot y}} \sqrt[5]{4 x^{4}}}{y}$

Step 9.2.2

Cancel the common factor.

$\frac{\sqrt[5]{\frac{y (8 x^{3})}{y \cdot y}} \sqrt[5]{4 x^{4}}}{y}$

Step 9.2.3

Rewrite the expression.

$\frac{\sqrt[5]{\frac{8 x^{3}}{y}} \sqrt[5]{4 x^{4}}}{y}$

Step 10

Simplify the numerator.

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Step 10.1

Combine using the product rule for radicals.

$\frac{\sqrt[5]{\frac{8 x^{3}}{y} (4 x^{4})}}{y}$

Step 10.2

Combine $4$ and $\frac{8 x^{3}}{y}$ .

$\frac{\sqrt[5]{\frac{4 (8 x^{3})}{y} x^{4}}}{y}$

Step 10.3

Combine $\frac{4 (8 x^{3})}{y}$ and $x^{4}$ .

$\frac{\sqrt[5]{\frac{4 (8 x^{3}) x^{4}}{y}}}{y}$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $4$ by $8$ .

$\frac{\sqrt[5]{\frac{32 x^{3} x^{4}}{y}}}{y}$

Step 11.2

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 11.2.1

Move $x^{4}$ .

$\frac{\sqrt[5]{\frac{32 (x^{4} x^{3})}{y}}}{y}$

Step 11.2.2

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

$\frac{\sqrt[5]{\frac{32 x^{4 + 3}}{y}}}{y}$

Step 11.2.3

Add $4$ and $3$ .

$\frac{\sqrt[5]{\frac{32 x^{7}}{y}}}{y}$

Step 12

Simplify the numerator.

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Step 12.1

Rewrite $\sqrt[5]{\frac{32 x^{7}}{y}}$ as $\frac{\sqrt[5]{32 x^{7}}}{\sqrt[5]{y}}$ .

$\frac{\frac{\sqrt[5]{32 x^{7}}}{\sqrt[5]{y}}}{y}$

Step 12.2

Simplify the numerator.

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Step 12.2.1

Rewrite $32 x^{7}$ as ${(2 x)}^{5} x^{2}$ .

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Step 12.2.1.1

Rewrite $32$ as $2^{5}$ .

$\frac{\frac{\sqrt[5]{2^{5} x^{7}}}{\sqrt[5]{y}}}{y}$

Step 12.2.1.2

Factor out $x^{5}$ .

$\frac{\frac{\sqrt[5]{2^{5} (x^{5} x^{2})}}{\sqrt[5]{y}}}{y}$

Step 12.2.1.3

Rewrite $2^{5} x^{5}$ as ${(2 x)}^{5}$ .

$\frac{\frac{\sqrt[5]{{(2 x)}^{5} x^{2}}}{\sqrt[5]{y}}}{y}$

Step 12.2.2

Pull terms out from under the radical.

$\frac{\frac{2 x \sqrt[5]{x^{2}}}{\sqrt[5]{y}}}{y}$

Step 12.3

Multiply $\frac{2 x \sqrt[5]{x^{2}}}{\sqrt[5]{y}}$ by $\frac{{\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{4}}$ .

$\frac{\frac{2 x \sqrt[5]{x^{2}}}{\sqrt[5]{y}} \cdot \frac{{\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{4}}}{y}$

Step 12.4

Combine and simplify the denominator.

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Step 12.4.1

Multiply $\frac{2 x \sqrt[5]{x^{2}}}{\sqrt[5]{y}}$ by $\frac{{\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{4}}$ .

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{\sqrt[5]{y} {\sqrt[5]{y}}^{4}}}{y}$

Step 12.4.2

Raise $\sqrt[5]{y}$ to the power of $1$ .

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{1} {\sqrt[5]{y}}^{4}}}{y}$

Step 12.4.3

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

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{1 + 4}}}{y}$

Step 12.4.4

Add $1$ and $4$ .

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{{\sqrt[5]{y}}^{5}}}{y}$

Step 12.4.5

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

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Step 12.4.5.1

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

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{{(y^{\frac{1}{5}})}^{5}}}{y}$

Step 12.4.5.2

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

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{y^{\frac{1}{5} \cdot 5}}}{y}$

Step 12.4.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{y^{\frac{5}{5}}}}{y}$

Step 12.4.5.4

Cancel the common factor of $5$ .

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Step 12.4.5.4.1

Cancel the common factor.

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{y^{\frac{5}{5}}}}{y}$

Step 12.4.5.4.2

Rewrite the expression.

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{y^{1}}}{y}$

Step 12.4.5.5

Simplify.

$\frac{\frac{2 x \sqrt[5]{x^{2}} {\sqrt[5]{y}}^{4}}{y}}{y}$

Step 12.5

Simplify the numerator.

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Step 12.5.1

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

$\frac{\frac{2 x \sqrt[5]{x^{2}} \sqrt[5]{y^{4}}}{y}}{y}$

Step 12.5.2

Combine using the product rule for radicals.

$\frac{\frac{2 x \sqrt[5]{y^{4} x^{2}}}{y}}{y}$

Step 13

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x \sqrt[5]{y^{4} x^{2}}}{y} \cdot \frac{1}{y}$

Step 14

Combine.

$\frac{2 x \sqrt[5]{y^{4} x^{2}} \cdot 1}{y \cdot y}$

Step 15

Simplify the expression.

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Step 15.1

Multiply $y$ by $y$ .

$\frac{2 x \sqrt[5]{y^{4} x^{2}} \cdot 1}{y^{2}}$

Step 15.2

Multiply $2 x \sqrt[5]{y^{4} x^{2}}$ by $1$ .

$\frac{2 x \sqrt[5]{y^{4} x^{2}}}{y^{2}}$