Algebra Examples

$\sqrt[3]{\frac{81 x^{2} y^{3}}{8 x y^{4} z}}$

Step 1

Reduce the expression $\frac{81 x^{2} y^{3}}{8 x y^{4} z}$ by cancelling the common factors.

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Step 1.1

Factor $x$ out of $81 x^{2} y^{3}$ .

$\sqrt[3]{\frac{x (81 x y^{3})}{8 x y^{4} z}}$

Step 1.2

Factor $x$ out of $8 x y^{4} z$ .

$\sqrt[3]{\frac{x (81 x y^{3})}{x (8 y^{4} z)}}$

Step 1.3

Cancel the common factor.

$\sqrt[3]{\frac{x (81 x y^{3})}{x (8 y^{4} z)}}$

Step 1.4

Rewrite the expression.

$\sqrt[3]{\frac{81 x y^{3}}{8 y^{4} z}}$

Step 2

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

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Step 2.1

Factor $y^{3}$ out of $81 x y^{3}$ .

$\sqrt[3]{\frac{y^{3} (81 x)}{8 y^{4} z}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $y^{3}$ out of $8 y^{4} z$ .

$\sqrt[3]{\frac{y^{3} (81 x)}{y^{3} (8 y z)}}$

Step 2.2.2

Cancel the common factor.

$\sqrt[3]{\frac{y^{3} (81 x)}{y^{3} (8 y z)}}$

Step 2.2.3

Rewrite the expression.

$\sqrt[3]{\frac{81 x}{8 y z}}$

Step 3

Rewrite $\frac{81 x}{8 y z}$ as ${(\frac{3}{2})}^{3} \frac{3 x}{y z}$ .

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Step 3.1

Factor the perfect power $3^{3}$ out of $81 x$ .

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

Step 3.2

Factor the perfect power $2^{3}$ out of $8 y z$ .

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

Step 3.3

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

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

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

Step 7

Combine fractions.

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Step 7.1

Combine and simplify the denominator.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.1.4

Add $1$ and $2$ .

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

Step 7.1.5

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

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Step 7.1.5.1

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

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

Step 7.1.5.2

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

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

Step 7.1.5.3

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

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

Step 7.1.5.4

Cancel the common factor of $3$ .

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Step 7.1.5.4.1

Cancel the common factor.

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

Step 7.1.5.4.2

Rewrite the expression.

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

Step 7.1.5.5

Simplify.

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

Step 7.2

Combine.

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

Apply the product rule to $y z$ .

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

Step 8.3

Combine using the product rule for radicals.

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

Step 9

Move $3$ to the left of $y^{2} z^{2}$ .

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