Algebra Examples

$\frac{\sqrt[3]{6 x^{2} y^{4}}}{2 \sqrt[3]{5 x^{7} y}}$

Step 1

Combine $\sqrt[3]{6 x^{2} y^{4}}$ and $\sqrt[3]{5 x^{7} y}$ into a single radical.

$\frac{\sqrt[3]{\frac{6 x^{2} y^{4}}{5 x^{7} y}}}{2}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x^{2}$ and $x^{7}$ .

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Step 2.1.1

Factor $x^{2}$ out of $6 x^{2} y^{4}$ .

$\frac{\sqrt[3]{\frac{x^{2} (6 y^{4})}{5 x^{7} y}}}{2}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $x^{2}$ out of $5 x^{7} y$ .

$\frac{\sqrt[3]{\frac{x^{2} (6 y^{4})}{x^{2} (5 x^{5} y)}}}{2}$

Step 2.1.2.2

Cancel the common factor.

$\frac{\sqrt[3]{\frac{x^{2} (6 y^{4})}{x^{2} (5 x^{5} y)}}}{2}$

Step 2.1.2.3

Rewrite the expression.

$\frac{\sqrt[3]{\frac{6 y^{4}}{5 x^{5} y}}}{2}$

Step 2.2

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

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Step 2.2.1

Factor $y$ out of $6 y^{4}$ .

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

$\frac{\sqrt[3]{\frac{6 y^{3}}{5 x^{5}}}}{2}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $\frac{6 y^{3}}{5 x^{5}}$ as ${(\frac{y}{x})}^{3} \frac{6}{5 x^{2}}$ .

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Step 3.1.1

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

$\frac{\sqrt[3]{\frac{y^{3} \cdot 6}{5 x^{5}}}}{2}$

Step 3.1.2

Factor the perfect power $x^{3}$ out of $5 x^{5}$ .

$\frac{\sqrt[3]{\frac{y^{3} \cdot 6}{x^{3} \cdot (5 x^{2})}}}{2}$

Step 3.1.3

Rearrange the fraction $\frac{y^{3} \cdot 6}{x^{3} \cdot (5 x^{2})}$ .

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

Step 3.2

Pull terms out from under the radical.

$\frac{\frac{y}{x} \sqrt[3]{\frac{6}{5 x^{2}}}}{2}$

Step 3.3

Rewrite $\sqrt[3]{\frac{6}{5 x^{2}}}$ as $\frac{\sqrt[3]{6}}{\sqrt[3]{5 x^{2}}}$ .

$\frac{\frac{y}{x} \cdot \frac{\sqrt[3]{6}}{\sqrt[3]{5 x^{2}}}}{2}$

Step 3.4

Combine.

$\frac{\frac{y \sqrt[3]{6}}{x \sqrt[3]{5 x^{2}}}}{2}$

Step 3.5

Multiply $\frac{y \sqrt[3]{6}}{x \sqrt[3]{5 x^{2}}}$ by $\frac{{\sqrt[3]{5 x^{2}}}^{2}}{{\sqrt[3]{5 x^{2}}}^{2}}$ .

$\frac{\frac{y \sqrt[3]{6}}{x \sqrt[3]{5 x^{2}}} \cdot \frac{{\sqrt[3]{5 x^{2}}}^{2}}{{\sqrt[3]{5 x^{2}}}^{2}}}{2}$

Step 3.6

Combine and simplify the denominator.

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Step 3.6.1

Multiply $\frac{y \sqrt[3]{6}}{x \sqrt[3]{5 x^{2}}}$ by $\frac{{\sqrt[3]{5 x^{2}}}^{2}}{{\sqrt[3]{5 x^{2}}}^{2}}$ .

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x \sqrt[3]{5 x^{2}} {\sqrt[3]{5 x^{2}}}^{2}}}{2}$

Step 3.6.2

Move $\sqrt[3]{5 x^{2}}$ .

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

Step 3.6.3

Raise $\sqrt[3]{5 x^{2}}$ to the power of $1$ .

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x ({\sqrt[3]{5 x^{2}}}^{1} {\sqrt[3]{5 x^{2}}}^{2})}}{2}$

Step 3.6.4

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

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x {\sqrt[3]{5 x^{2}}}^{1 + 2}}}{2}$

Step 3.6.5

Add $1$ and $2$ .

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x {\sqrt[3]{5 x^{2}}}^{3}}}{2}$

Step 3.6.6

Rewrite ${\sqrt[3]{5 x^{2}}}^{3}$ as $5 x^{2}$ .

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Step 3.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 x^{2}}$ as ${(5 x^{2})}^{\frac{1}{3}}$ .

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x {({(5 x^{2})}^{\frac{1}{3}})}^{3}}}{2}$

Step 3.6.6.2

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

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x {(5 x^{2})}^{\frac{1}{3} \cdot 3}}}{2}$

Step 3.6.6.3

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

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

Step 3.6.6.4

Cancel the common factor of $3$ .

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Step 3.6.6.4.1

Cancel the common factor.

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

Step 3.6.6.4.2

Rewrite the expression.

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x {(5 x^{2})}^{1}}}{2}$

Step 3.6.6.5

Simplify.

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

Step 3.7

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.7.1

Move $x^{2}$ .

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x^{2} x \cdot 5}}{2}$

Step 3.7.2

Multiply $x^{2}$ by $x$ .

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Step 3.7.2.1

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

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x^{2} x^{1} \cdot 5}}{2}$

Step 3.7.2.2

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

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x^{2 + 1} \cdot 5}}{2}$

Step 3.7.3

Add $2$ and $1$ .

$\frac{\frac{y \sqrt[3]{6} {\sqrt[3]{5 x^{2}}}^{2}}{x^{3} \cdot 5}}{2}$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Rewrite ${\sqrt[3]{5 x^{2}}}^{2}$ as $\sqrt[3]{{(5 x^{2})}^{2}}$ .

$\frac{\frac{y \sqrt[3]{6} \sqrt[3]{{(5 x^{2})}^{2}}}{x^{3} \cdot 5}}{2}$

Step 3.8.2

Apply the product rule to $5 x^{2}$ .

$\frac{\frac{y \sqrt[3]{6} \sqrt[3]{5^{2} {(x^{2})}^{2}}}{x^{3} \cdot 5}}{2}$

Step 3.8.3

Raise $5$ to the power of $2$ .

$\frac{\frac{y \sqrt[3]{6} \sqrt[3]{25 {(x^{2})}^{2}}}{x^{3} \cdot 5}}{2}$

Step 3.8.4

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

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Step 3.8.4.1

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

$\frac{\frac{y \sqrt[3]{6} \sqrt[3]{25 x^{2 \cdot 2}}}{x^{3} \cdot 5}}{2}$

Step 3.8.4.2

Multiply $2$ by $2$ .

$\frac{\frac{y \sqrt[3]{6} \sqrt[3]{25 x^{4}}}{x^{3} \cdot 5}}{2}$

Step 3.8.5

Rewrite $25 x^{4}$ as $x^{3} \cdot (25 x)$ .

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Step 3.8.5.1

Factor out $x^{3}$ .

$\frac{\frac{y \sqrt[3]{6} \sqrt[3]{25 (x^{3} x)}}{x^{3} \cdot 5}}{2}$

Step 3.8.5.2

Reorder $25$ and $x^{3}$ .

$\frac{\frac{y \sqrt[3]{6} \sqrt[3]{x^{3} \cdot 25 x}}{x^{3} \cdot 5}}{2}$

Step 3.8.5.3

Add parentheses.

$\frac{\frac{y \sqrt[3]{6} \sqrt[3]{x^{3} \cdot (25 x)}}{x^{3} \cdot 5}}{2}$

Step 3.8.6

Pull terms out from under the radical.

$\frac{\frac{y \sqrt[3]{6} x \sqrt[3]{25 x}}{x^{3} \cdot 5}}{2}$

Step 3.8.7

Combine exponents.

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Step 3.8.7.1

Combine using the product rule for radicals.

$\frac{\frac{y x \sqrt[3]{6 (25 x)}}{x^{3} \cdot 5}}{2}$

Step 3.8.7.2

Multiply $6$ by $25$ .

$\frac{\frac{y x \sqrt[3]{150 x}}{x^{3} \cdot 5}}{2}$

Step 3.9

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

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Step 3.9.1

Factor $x$ out of $y x \sqrt[3]{150 x}$ .

$\frac{\frac{x (y \sqrt[3]{150 x})}{x^{3} \cdot 5}}{2}$

Step 3.9.2

Cancel the common factors.

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Step 3.9.2.1

Factor $x$ out of $x^{3} \cdot 5$ .

$\frac{\frac{x (y \sqrt[3]{150 x})}{x (x^{2} \cdot 5)}}{2}$

Step 3.9.2.2

Cancel the common factor.

$\frac{\frac{x (y \sqrt[3]{150 x})}{x (x^{2} \cdot 5)}}{2}$

Step 3.9.2.3

Rewrite the expression.

$\frac{\frac{y \sqrt[3]{150 x}}{x^{2} \cdot 5}}{2}$

Step 3.10

Move $5$ to the left of $x^{2}$ .

$\frac{\frac{y \sqrt[3]{150 x}}{5 x^{2}}}{2}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{y \sqrt[3]{150 x}}{5 x^{2}} \cdot \frac{1}{2}$

Step 5

Combine.

$\frac{y \sqrt[3]{150 x} \cdot 1}{5 x^{2} \cdot 2}$

Step 6

Multiply.

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Step 6.1

Multiply $y$ by $1$ .

$\frac{y \sqrt[3]{150 x}}{5 x^{2} \cdot 2}$

Step 6.2

Multiply $2$ by $5$ .

$\frac{y \sqrt[3]{150 x}}{10 x^{2}}$