Algebra Examples

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

Step 1

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

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

Step 2

Reduce the expression $\frac{x^{3} y^{2} z^{7}}{x y^{4}}$ by cancelling the common factors.

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Step 2.1

Factor $x$ out of $x^{3} y^{2} z^{7}$ .

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

Step 2.2

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

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

Step 2.3

Cancel the common factor.

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

Step 2.4

Rewrite the expression.

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

Step 3

Cancel the common factor of $y^{2}$ and $y^{4}$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Rewrite $\sqrt[3]{\frac{x^{2} z^{7}}{y^{2}}}$ as $\frac{\sqrt[3]{x^{2} z^{7}}}{\sqrt[3]{y^{2}}}$ .

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $x^{2} z^{7}$ as ${(z^{2})}^{3} (x^{2} z)$ .

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Step 5.1.1

Factor out $z^{6}$ .

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

Step 5.1.2

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

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

Step 5.1.3

Reorder $x^{2}$ and ${(z^{2})}^{3}$ .

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

Step 5.1.4

Add parentheses.

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

Step 5.2

Pull terms out from under the radical.

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $2$ .

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

Step 7.5

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

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Step 7.5.1

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

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

Step 7.5.2

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

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

Step 7.5.3

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

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

Step 7.5.4

Multiply $2$ by $3$ .

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

Step 7.5.5

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

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Step 7.5.5.1

Factor $3$ out of $6$ .

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

Step 7.5.5.2

Cancel the common factors.

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Step 7.5.5.2.1

Factor $3$ out of $3$ .

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

Step 7.5.5.2.2

Cancel the common factor.

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

Step 7.5.5.2.3

Rewrite the expression.

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

Step 7.5.5.2.4

Divide $2$ by $1$ .

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Multiply $2$ by $2$ .

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

Step 8.3

Factor out $y^{3}$ .

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

Step 8.4

Pull terms out from under the radical.

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

Step 8.5

Combine using the product rule for radicals.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

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

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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