Algebra Examples

$\sqrt{\frac{a^{4} b^{4}}{c^{3}}}$

Step 1

Rewrite $\frac{a^{4} b^{4}}{c^{3}}$ as ${(\frac{a^{2} b^{2}}{c})}^{2} \frac{1}{c}$ .

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Step 1.1

Factor the perfect power ${(a^{2} b^{2})}^{2}$ out of $a^{4} b^{4}$ .

$\sqrt{\frac{{(a^{2} b^{2})}^{2} \cdot 1}{c^{3}}}$

Step 1.2

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

$\sqrt{\frac{{(a^{2} b^{2})}^{2} \cdot 1}{c^{2} c}}$

Step 1.3

Rearrange the fraction $\frac{{(a^{2} b^{2})}^{2} \cdot 1}{c^{2} c}$ .

$\sqrt{{(\frac{a^{2} b^{2}}{c})}^{2} \frac{1}{c}}$

Step 2

Pull terms out from under the radical.

$| \frac{a^{2} b^{2}}{c} | \sqrt{\frac{1}{c}}$

Step 3

Remove non-negative terms from the absolute value.

$\frac{a^{2} b^{2}}{| c |} \sqrt{\frac{1}{c}}$

Step 4

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

$\frac{a^{2} b^{2}}{| c |} \cdot \frac{\sqrt{1}}{\sqrt{c}}$

Step 5

Combine.

$\frac{a^{2} b^{2} \sqrt{1}}{| c | \sqrt{c}}$

Step 6

Simplify the numerator.

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Step 6.1

Any root of $1$ is $1$ .

$\frac{a^{2} b^{2} \cdot 1}{| c | \sqrt{c}}$

Step 6.2

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

$\frac{a^{2} b^{2}}{| c | \sqrt{c}}$

Step 7

Multiply $\frac{a^{2} b^{2}}{| c | \sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$\frac{a^{2} b^{2}}{| c | \sqrt{c}} \cdot \frac{\sqrt{c}}{\sqrt{c}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{a^{2} b^{2}}{| c | \sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$\frac{a^{2} b^{2} \sqrt{c}}{| c | \sqrt{c} \sqrt{c}}$

Step 8.2

Move $\sqrt{c}$ .

$\frac{a^{2} b^{2} \sqrt{c}}{| c | (\sqrt{c} \sqrt{c})}$

Step 8.3

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

$\frac{a^{2} b^{2} \sqrt{c}}{| c | ({\sqrt{c}}^{1} \sqrt{c})}$

Step 8.4

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

$\frac{a^{2} b^{2} \sqrt{c}}{| c | ({\sqrt{c}}^{1} {\sqrt{c}}^{1})}$

Step 8.5

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

$\frac{a^{2} b^{2} \sqrt{c}}{| c | {\sqrt{c}}^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$\frac{a^{2} b^{2} \sqrt{c}}{| c | {\sqrt{c}}^{2}}$

Step 8.7

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

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Step 8.7.1

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

$\frac{a^{2} b^{2} \sqrt{c}}{| c | {(c^{\frac{1}{2}})}^{2}}$

Step 8.7.2

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

$\frac{a^{2} b^{2} \sqrt{c}}{| c | c^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

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

$\frac{a^{2} b^{2} \sqrt{c}}{| c | c^{\frac{2}{2}}}$

Step 8.7.4

Cancel the common factor of $2$ .

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Step 8.7.4.1

Cancel the common factor.

$\frac{a^{2} b^{2} \sqrt{c}}{| c | c^{\frac{2}{2}}}$

Step 8.7.4.2

Rewrite the expression.

$\frac{a^{2} b^{2} \sqrt{c}}{| c | c^{1}}$

Step 8.7.5

Simplify.

$\frac{a^{2} b^{2} \sqrt{c}}{| c | c}$

Step 9

Reorder factors in $\frac{a^{2} b^{2} \sqrt{c}}{| c | c}$ .

$\frac{a^{2} b^{2} \sqrt{c}}{c | c |}$