Pre-Algebra Examples

$\sqrt{\frac{128 x^{3} y}{27 x y^{6} z^{9}}}$

Step 1

Reduce the expression $\frac{128 x^{3} y}{27 x y^{6} z^{9}}$ by cancelling the common factors.

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Step 1.1

Factor $x$ out of $128 x^{3} y$ .

$\sqrt{\frac{x (128 x^{2} y)}{27 x y^{6} z^{9}}}$

Step 1.2

Factor $x$ out of $27 x y^{6} z^{9}$ .

$\sqrt{\frac{x (128 x^{2} y)}{x (27 y^{6} z^{9})}}$

Step 1.3

Cancel the common factor.

$\sqrt{\frac{x (128 x^{2} y)}{x (27 y^{6} z^{9})}}$

Step 1.4

Rewrite the expression.

$\sqrt{\frac{128 x^{2} y}{27 y^{6} z^{9}}}$

Step 2

Cancel the common factor of $y$ and $y^{6}$ .

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Step 2.1

Factor $y$ out of $128 x^{2} y$ .

$\sqrt{\frac{y (128 x^{2})}{27 y^{6} z^{9}}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $y$ out of $27 y^{6} z^{9}$ .

$\sqrt{\frac{y (128 x^{2})}{y (27 y^{5} z^{9})}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{y (128 x^{2})}{y (27 y^{5} z^{9})}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{128 x^{2}}{27 y^{5} z^{9}}}$

Step 3

Rewrite $\frac{128 x^{2}}{27 y^{5} z^{9}}$ as ${(\frac{8 x}{3 y^{2} z^{4}})}^{2} \frac{2}{3 y z}$ .

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Step 3.1

Factor the perfect power ${(8 x)}^{2}$ out of $128 x^{2}$ .

$\sqrt{\frac{{(8 x)}^{2} \cdot 2}{27 y^{5} z^{9}}}$

Step 3.2

Rearrange the fraction $\frac{{(8 x)}^{2} \cdot 2}{{(3 y^{2} z^{4})}^{2} \cdot (3 y z)}$ .

$\sqrt{{(\frac{8 x}{3 y^{2} z^{4}})}^{2} \frac{2}{3 y z}}$

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite $\sqrt{\frac{2}{3 y z}}$ as $\frac{\sqrt{2}}{\sqrt{3 y z}}$ .

$\frac{8 x}{3 y^{2} z^{4}} \cdot \frac{\sqrt{2}}{\sqrt{3 y z}}$

Step 6

Combine.

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

Step 7

Multiply $\frac{8 x \sqrt{2}}{3 y^{2} z^{4} \sqrt{3 y z}}$ by $\frac{\sqrt{3 y z}}{\sqrt{3 y z}}$ .

$\frac{8 x \sqrt{2}}{3 y^{2} z^{4} \sqrt{3 y z}} \cdot \frac{\sqrt{3 y z}}{\sqrt{3 y z}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{8 x \sqrt{2}}{3 y^{2} z^{4} \sqrt{3 y z}}$ by $\frac{\sqrt{3 y z}}{\sqrt{3 y z}}$ .

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

Step 8.2

Move $\sqrt{3 y z}$ .

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} (\sqrt{3 y z} \sqrt{3 y z})}$

Step 8.3

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} ({\sqrt{3 y z}}^{1} \sqrt{3 y z})}$

Step 8.4

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} ({\sqrt{3 y z}}^{1} {\sqrt{3 y z}}^{1})}$

Step 8.5

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} {\sqrt{3 y z}}^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

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

Step 8.7

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

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Step 8.7.1

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} {({(3 y z)}^{\frac{1}{2}})}^{2}}$

Step 8.7.2

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} {(3 y z)}^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} {(3 y z)}^{\frac{2}{2}}}$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} {(3 y z)}^{\frac{2}{2}}}$

Step 8.7.4.2

Rewrite the expression.

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} {(3 y z)}^{1}}$

Step 8.7.5

Simplify.

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{2} z^{4} (3 y z)}$

Step 9

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

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Step 9.1

Move $y$ .

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 (y \cdot y^{2}) z^{4} (3 z)}$

Step 9.2

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

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Step 9.2.1

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 (y^{1} y^{2}) z^{4} (3 z)}$

Step 9.2.2

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{1 + 2} z^{4} (3 z)}$

Step 9.3

Add $1$ and $2$ .

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{3} z^{4} (3 z)}$

Step 10

Multiply $z^{4}$ by $z$ by adding the exponents.

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Step 10.1

Move $z$ .

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{3} (z \cdot z^{4}) \cdot 3}$

Step 10.2

Multiply $z$ by $z^{4}$ .

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Step 10.2.1

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{3} (z^{1} z^{4}) \cdot 3}$

Step 10.2.2

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

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{3} z^{1 + 4} \cdot 3}$

Step 10.3

Add $1$ and $4$ .

$\frac{8 x \sqrt{2} \sqrt{3 y z}}{3 y^{3} z^{5} \cdot 3}$

Step 11

Simplify the numerator.

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Step 11.1

Combine using the product rule for radicals.

$\frac{8 x \sqrt{3 y z \cdot 2}}{3 y^{3} z^{5} \cdot 3}$

Step 11.2

Multiply $2$ by $3$ .

$\frac{8 x \sqrt{6 y z}}{3 y^{3} z^{5} \cdot 3}$

Step 12

Multiply $3$ by $3$ .

$\frac{8 x \sqrt{6 y z}}{9 y^{3} z^{5}}$