Algebra Examples

$\sqrt{\frac{a^{2} b}{c^{7} d^{4}}}$

Step 1

Rewrite $\frac{a^{2} b}{c^{7} d^{4}}$ as ${(\frac{a}{c^{3} d^{2}})}^{2} \frac{b}{c}$ .

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Step 1.1

Factor the perfect power ${(c^{3} d^{2})}^{2}$ out of $c^{7} d^{4}$ .

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

Step 1.2

Rearrange the fraction $\frac{a^{2} b}{{(c^{3} d^{2})}^{2} c}$ .

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

Step 2

Pull terms out from under the radical.

$\frac{a}{c^{3} d^{2}} \sqrt{\frac{b}{c}}$

Step 3

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

$\frac{a}{c^{3} d^{2}} \cdot \frac{\sqrt{b}}{\sqrt{c}}$

Step 4

Combine.

$\frac{a \sqrt{b}}{c^{3} d^{2} \sqrt{c}}$

Step 5

Multiply $\frac{a \sqrt{b}}{c^{3} d^{2} \sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$\frac{a \sqrt{b}}{c^{3} d^{2} \sqrt{c}} \cdot \frac{\sqrt{c}}{\sqrt{c}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{a \sqrt{b}}{c^{3} d^{2} \sqrt{c}}$ by $\frac{\sqrt{c}}{\sqrt{c}}$ .

$\frac{a \sqrt{b} \sqrt{c}}{c^{3} d^{2} \sqrt{c} \sqrt{c}}$

Step 6.2

Move $\sqrt{c}$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

$\frac{a \sqrt{b} \sqrt{c}}{c^{3} d^{2} {\sqrt{c}}^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$\frac{a \sqrt{b} \sqrt{c}}{c^{3} d^{2} {\sqrt{c}}^{2}}$

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

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

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

Step 6.7.3

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

$\frac{a \sqrt{b} \sqrt{c}}{c^{3} d^{2} c^{\frac{2}{2}}}$

Step 6.7.4

Cancel the common factor of $2$ .

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Step 6.7.4.1

Cancel the common factor.

$\frac{a \sqrt{b} \sqrt{c}}{c^{3} d^{2} c^{\frac{2}{2}}}$

Step 6.7.4.2

Rewrite the expression.

$\frac{a \sqrt{b} \sqrt{c}}{c^{3} d^{2} c^{1}}$

Step 6.7.5

Simplify.

$\frac{a \sqrt{b} \sqrt{c}}{c^{3} d^{2} c}$

Step 7

Multiply $c^{3}$ by $c$ by adding the exponents.

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Step 7.1

Move $c$ .

$\frac{a \sqrt{b} \sqrt{c}}{c \cdot c^{3} d^{2}}$

Step 7.2

Multiply $c$ by $c^{3}$ .

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Step 7.2.1

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

$\frac{a \sqrt{b} \sqrt{c}}{c^{1} c^{3} d^{2}}$

Step 7.2.2

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

$\frac{a \sqrt{b} \sqrt{c}}{c^{1 + 3} d^{2}}$

Step 7.3

Add $1$ and $3$ .

$\frac{a \sqrt{b} \sqrt{c}}{c^{4} d^{2}}$

Step 8

Combine using the product rule for radicals.

$\frac{a \sqrt{c b}}{c^{4} d^{2}}$