Algebra Examples

$\frac{\sqrt[3]{48 x^{3} y^{7}}}{\sqrt[3]{8 x^{5} y}}$

Step 1

Combine $\sqrt[3]{48 x^{3} y^{7}}$ and $\sqrt[3]{8 x^{5} y}$ into a single radical.

$\sqrt[3]{\frac{48 x^{3} y^{7}}{8 x^{5} y}}$

Step 2

Reduce the expression $\frac{48 x^{3} y^{7}}{8 x^{5} y}$ by cancelling the common factors.

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Step 2.1

Factor $8$ out of $48 x^{3} y^{7}$ .

$\sqrt[3]{\frac{8 (6 x^{3} y^{7})}{8 x^{5} y}}$

Step 2.2

Factor $8$ out of $8 x^{5} y$ .

$\sqrt[3]{\frac{8 (6 x^{3} y^{7})}{8 (x^{5} y)}}$

Step 2.3

Cancel the common factor.

$\sqrt[3]{\frac{8 (6 x^{3} y^{7})}{8 (x^{5} y)}}$

Step 2.4

Rewrite the expression.

$\sqrt[3]{\frac{6 x^{3} y^{7}}{x^{5} y}}$

Step 3

Cancel the common factor of $x^{3}$ and $x^{5}$ .

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Step 3.1

Factor $x^{3}$ out of $6 x^{3} y^{7}$ .

$\sqrt[3]{\frac{x^{3} (6 y^{7})}{x^{5} y}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x^{3}$ out of $x^{5} y$ .

$\sqrt[3]{\frac{x^{3} (6 y^{7})}{x^{3} (x^{2} y)}}$

Step 3.2.2

Cancel the common factor.

$\sqrt[3]{\frac{x^{3} (6 y^{7})}{x^{3} (x^{2} y)}}$

Step 3.2.3

Rewrite the expression.

$\sqrt[3]{\frac{6 y^{7}}{x^{2} y}}$

Step 4

Cancel the common factor of $y^{7}$ and $y$ .

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Step 4.1

Factor $y$ out of $6 y^{7}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $6 y^{6}$ as ${(y^{2})}^{3} \cdot 6$ .

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Step 6.1.1

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

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

Step 6.1.2

Reorder $6$ and ${(y^{2})}^{3}$ .

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

Step 6.2

Pull terms out from under the radical.

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

$\frac{y^{2} \sqrt[3]{6} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1} {\sqrt[3]{x^{2}}}^{2}}$

Step 8.3

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

$\frac{y^{2} \sqrt[3]{6} {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1 + 2}}$

Step 8.4

Add $1$ and $2$ .

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

Step 8.5

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

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Step 8.5.1

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

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

Step 8.5.2

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

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

Step 8.5.3

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

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

Step 8.5.4

Multiply $2$ by $3$ .

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

Step 8.5.5

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

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Step 8.5.5.1

Factor $3$ out of $6$ .

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

Step 8.5.5.2

Cancel the common factors.

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Step 8.5.5.2.1

Factor $3$ out of $3$ .

$\frac{y^{2} \sqrt[3]{6} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 (1)}}}$

Step 8.5.5.2.2

Cancel the common factor.

$\frac{y^{2} \sqrt[3]{6} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Step 8.5.5.2.3

Rewrite the expression.

$\frac{y^{2} \sqrt[3]{6} {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{1}}}$

Step 8.5.5.2.4

Divide $2$ by $1$ .

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

Step 9

Simplify the numerator.

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

Multiply $2$ by $2$ .

$\frac{y^{2} \sqrt[3]{6} \sqrt[3]{x^{4}}}{x^{2}}$

Step 9.3

Factor out $x^{3}$ .

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

Step 9.4

Pull terms out from under the radical.

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

Step 9.5

Combine using the product rule for radicals.

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

Step 10

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

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Step 10.1

Factor $x$ out of $y^{2} x \sqrt[3]{6 x}$ .

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

Step 10.2

Cancel the common factors.

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Step 10.2.1

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

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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