Algebra Examples

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

Step 1

Apply the product rule to $x^{5} y^{- 8}$ .

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

Step 2

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 2.1

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

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

Step 2.2

Multiply $5$ by $2$ .

$\sqrt{\sqrt[3]{x^{10} {(y^{- 8})}^{2}}}$

Step 3

Multiply the exponents in ${(y^{- 8})}^{2}$ .

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Step 3.1

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

$\sqrt{\sqrt[3]{x^{10} y^{- 8 \cdot 2}}}$

Step 3.2

Multiply $- 8$ by $2$ .

$\sqrt{\sqrt[3]{x^{10} y^{- 16}}}$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt{\sqrt[3]{x^{10} \frac{1}{y^{16}}}}$

Step 5

Combine $x^{10}$ and $\frac{1}{y^{16}}$ .

$\sqrt{\sqrt[3]{\frac{x^{10}}{y^{16}}}}$

Step 6

Rewrite $\frac{x^{10}}{y^{16}}$ as ${(\frac{x^{3}}{y^{5}})}^{3} \frac{x}{y}$ .

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Step 6.1

Factor the perfect power ${(x^{3})}^{3}$ out of $x^{10}$ .

$\sqrt{\sqrt[3]{\frac{{(x^{3})}^{3} x}{y^{16}}}}$

Step 6.2

Factor the perfect power ${(y^{5})}^{3}$ out of $y^{16}$ .

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

Step 6.3

Rearrange the fraction $\frac{{(x^{3})}^{3} x}{{(y^{5})}^{3} y}$ .

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

Step 7

Pull terms out from under the radical.

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

Step 8

Rewrite $\sqrt[3]{\frac{x}{y}}$ as $\frac{\sqrt[3]{x}}{\sqrt[3]{y}}$ .

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

Step 9

Combine.

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

Step 10

Multiply $\frac{x^{3} \sqrt[3]{x}}{y^{5} \sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

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

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{x^{3} \sqrt[3]{x}}{y^{5} \sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

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

Step 11.2

Move $\sqrt[3]{y}$ .

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

Step 11.3

Raise $\sqrt[3]{y}$ to the power of $1$ .

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

Step 11.4

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

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

Step 11.5

Add $1$ and $2$ .

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

Step 11.6

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

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Step 11.6.1

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

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

Step 11.6.2

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

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

Step 11.6.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 11.6.4

Cancel the common factor of $3$ .

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Step 11.6.4.1

Cancel the common factor.

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

Step 11.6.4.2

Rewrite the expression.

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

Step 11.6.5

Simplify.

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

Step 12

Multiply $y^{5}$ by $y$ by adding the exponents.

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Step 12.1

Multiply $y^{5}$ by $y$ .

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Step 12.1.1

Raise $y$ to the power of $1$ .

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

Step 12.1.2

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

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

Step 12.2

Add $5$ and $1$ .

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

Step 13

Simplify the numerator.

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Step 13.1

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

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

Step 13.2

Combine using the product rule for radicals.

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

Step 14

Rewrite $\frac{x^{3} \sqrt[3]{y^{2} x}}{y^{6}}$ as ${(\frac{x}{y^{3}})}^{2} (x \sqrt[3]{y^{2} x})$ .

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Step 14.1

Factor the perfect power $x^{2}$ out of $x^{3} \sqrt[3]{y^{2} x}$ .

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

Step 14.2

Factor the perfect power ${(y^{3})}^{2}$ out of $y^{6}$ .

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

Step 14.3

Rearrange the fraction $\frac{x^{2} (x \sqrt[3]{y^{2} x})}{{(y^{3})}^{2} \cdot 1}$ .

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

Step 15

Pull terms out from under the radical.

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

Step 16

Combine $\frac{x}{y^{3}}$ and $\sqrt{x \sqrt[3]{y^{2} x}}$ .

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