Algebra Examples

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

Step 1

Rewrite $\frac{x^{4} y^{4}}{z^{3}}$ as ${(\frac{x^{2} y^{2}}{z})}^{2} \frac{1}{z}$ .

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Step 1.1

Factor the perfect power ${(x^{2} y^{2})}^{2}$ out of $x^{4} y^{4}$ .

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

Step 1.2

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

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

Step 1.3

Rearrange the fraction $\frac{{(x^{2} y^{2})}^{2} \cdot 1}{z^{2} z}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $\sqrt{\frac{1}{z}}$ as $\frac{\sqrt{1}}{\sqrt{z}}$ .

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

Step 4

Combine.

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

Step 5

Simplify the numerator.

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Step 5.1

Any root of $1$ is $1$ .

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

Step 5.2

Multiply $x^{2}$ by $1$ .

$\frac{x^{2} y^{2}}{z \sqrt{z}}$

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

$\frac{x^{2} y^{2} \sqrt{z}}{z \sqrt{z} \sqrt{z}}$

Step 7.2

Move $\sqrt{z}$ .

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

Add $1$ and $1$ .

$\frac{x^{2} y^{2} \sqrt{z}}{z {\sqrt{z}}^{2}}$

Step 7.7

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

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Step 7.7.1

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

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

Step 7.7.2

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

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

Step 7.7.3

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

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

Step 7.7.4

Cancel the common factor of $2$ .

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Step 7.7.4.1

Cancel the common factor.

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

Step 7.7.4.2

Rewrite the expression.

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

Step 7.7.5

Simplify.

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

Step 8

Multiply $z$ by $z$ .

$\frac{x^{2} y^{2} \sqrt{z}}{z^{2}}$