Algebra Examples

$\sqrt{\frac{16 a^{2} b^{9} c^{3}}{36 a^{4} b^{5} c^{6}}}$

Step 1

Reduce the expression $\frac{16 a^{2} b^{9} c^{3}}{36 a^{4} b^{5} c^{6}}$ by cancelling the common factors.

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Step 1.1

Factor $4$ out of $16 a^{2} b^{9} c^{3}$ .

$\sqrt{\frac{4 (4 a^{2} b^{9} c^{3})}{36 a^{4} b^{5} c^{6}}}$

Step 1.2

Factor $4$ out of $36 a^{4} b^{5} c^{6}$ .

$\sqrt{\frac{4 (4 a^{2} b^{9} c^{3})}{4 (9 a^{4} b^{5} c^{6})}}$

Step 1.3

Cancel the common factor.

$\sqrt{\frac{4 (4 a^{2} b^{9} c^{3})}{4 (9 a^{4} b^{5} c^{6})}}$

Step 1.4

Rewrite the expression.

$\sqrt{\frac{4 a^{2} b^{9} c^{3}}{9 a^{4} b^{5} c^{6}}}$

Step 2

Cancel the common factor of $a^{2}$ and $a^{4}$ .

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Step 2.1

Factor $a^{2}$ out of $4 a^{2} b^{9} c^{3}$ .

$\sqrt{\frac{a^{2} (4 b^{9} c^{3})}{9 a^{4} b^{5} c^{6}}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $a^{2}$ out of $9 a^{4} b^{5} c^{6}$ .

$\sqrt{\frac{a^{2} (4 b^{9} c^{3})}{a^{2} (9 a^{2} b^{5} c^{6})}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{a^{2} (4 b^{9} c^{3})}{a^{2} (9 a^{2} b^{5} c^{6})}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{4 b^{9} c^{3}}{9 a^{2} b^{5} c^{6}}}$

Step 3

Cancel the common factor of $b^{9}$ and $b^{5}$ .

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Step 3.1

Factor $b^{5}$ out of $4 b^{9} c^{3}$ .

$\sqrt{\frac{b^{5} (4 b^{4} c^{3})}{9 a^{2} b^{5} c^{6}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $b^{5}$ out of $9 a^{2} b^{5} c^{6}$ .

$\sqrt{\frac{b^{5} (4 b^{4} c^{3})}{b^{5} (9 a^{2} c^{6})}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{b^{5} (4 b^{4} c^{3})}{b^{5} (9 a^{2} c^{6})}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{4 b^{4} c^{3}}{9 a^{2} c^{6}}}$

Step 4

Cancel the common factor of $c^{3}$ and $c^{6}$ .

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Step 4.1

Factor $c^{3}$ out of $4 b^{4} c^{3}$ .

$\sqrt{\frac{c^{3} (4 b^{4})}{9 a^{2} c^{6}}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $c^{3}$ out of $9 a^{2} c^{6}$ .

$\sqrt{\frac{c^{3} (4 b^{4})}{c^{3} (9 a^{2} c^{3})}}$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{c^{3} (4 b^{4})}{c^{3} (9 a^{2} c^{3})}}$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{4 b^{4}}{9 a^{2} c^{3}}}$

Step 5

Rewrite $\frac{4 b^{4}}{9 a^{2} c^{3}}$ as ${(\frac{2 b^{2}}{3 a c})}^{2} \frac{1}{c}$ .

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Step 5.1

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

$\sqrt{\frac{{(2 b^{2})}^{2} \cdot 1}{9 a^{2} c^{3}}}$

Step 5.2

Factor the perfect power ${(3 a c)}^{2}$ out of $9 a^{2} c^{3}$ .

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

Step 5.3

Rearrange the fraction $\frac{{(2 b^{2})}^{2} \cdot 1}{{(3 a c)}^{2} c}$ .

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

Step 6

Pull terms out from under the radical.

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

Step 7

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

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

Step 8

Combine.

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

Step 9

Simplify the numerator.

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Step 9.1

Any root of $1$ is $1$ .

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

Step 9.2

Multiply $2$ by $1$ .

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

Step 10

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

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

Step 11

Combine and simplify the denominator.

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Step 11.1

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

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

Step 11.2

Move $\sqrt{c}$ .

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

Step 11.3

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

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

Step 11.4

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

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

Step 11.5

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

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

Step 11.6

Add $1$ and $1$ .

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

Step 11.7

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

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Step 11.7.1

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

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

Step 11.7.2

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

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

Step 11.7.3

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

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

Step 11.7.4

Cancel the common factor of $2$ .

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Step 11.7.4.1

Cancel the common factor.

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

Step 11.7.4.2

Rewrite the expression.

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

Step 11.7.5

Simplify.

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

Step 12

Multiply $c$ by $c$ by adding the exponents.

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Step 12.1

Move $c$ .

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

Step 12.2

Multiply $c$ by $c$ .

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