Precalculus Examples

$\sqrt{\frac{441 a b^{6}}{108 a^{5} b}}$

Step 1

Reduce the expression $\frac{441 a b^{6}}{108 a^{5} b}$ by cancelling the common factors.

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Step 1.1

Factor $9$ out of $441 a b^{6}$ .

$\sqrt{\frac{9 (49 a b^{6})}{108 a^{5} b}}$

Step 1.2

Factor $9$ out of $108 a^{5} b$ .

$\sqrt{\frac{9 (49 a b^{6})}{9 (12 a^{5} b)}}$

Step 1.3

Cancel the common factor.

$\sqrt{\frac{9 (49 a b^{6})}{9 (12 a^{5} b)}}$

Step 1.4

Rewrite the expression.

$\sqrt{\frac{49 a b^{6}}{12 a^{5} b}}$

Step 2

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

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Step 2.1

Factor $a$ out of $49 a b^{6}$ .

$\sqrt{\frac{a (49 b^{6})}{12 a^{5} b}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $a$ out of $12 a^{5} b$ .

$\sqrt{\frac{a (49 b^{6})}{a (12 a^{4} b)}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{a (49 b^{6})}{a (12 a^{4} b)}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{49 b^{6}}{12 a^{4} b}}$

Step 3

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

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Step 3.1

Factor $b$ out of $49 b^{6}$ .

$\sqrt{\frac{b (49 b^{5})}{12 a^{4} b}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $b$ out of $12 a^{4} b$ .

$\sqrt{\frac{b (49 b^{5})}{b (12 a^{4})}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{b (49 b^{5})}{b (12 a^{4})}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{49 b^{5}}{12 a^{4}}}$

Step 4

Rewrite $\frac{49 b^{5}}{12 a^{4}}$ as ${(\frac{7 b^{2}}{2 a^{2}})}^{2} \frac{b}{3}$ .

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Step 4.1

Factor the perfect power ${(7 b^{2})}^{2}$ out of $49 b^{5}$ .

$\sqrt{\frac{{(7 b^{2})}^{2} b}{12 a^{4}}}$

Step 4.2

Factor the perfect power ${(2 a^{2})}^{2}$ out of $12 a^{4}$ .

$\sqrt{\frac{{(7 b^{2})}^{2} b}{{(2 a^{2})}^{2} \cdot 3}}$

Step 4.3

Rearrange the fraction $\frac{{(7 b^{2})}^{2} b}{{(2 a^{2})}^{2} \cdot 3}$ .

$\sqrt{{(\frac{7 b^{2}}{2 a^{2}})}^{2} \frac{b}{3}}$

Step 5

Pull terms out from under the radical.

$\frac{7 b^{2}}{2 a^{2}} \sqrt{\frac{b}{3}}$

Step 6

Rewrite $\sqrt{\frac{b}{3}}$ as $\frac{\sqrt{b}}{\sqrt{3}}$ .

$\frac{7 b^{2}}{2 a^{2}} \cdot \frac{\sqrt{b}}{\sqrt{3}}$

Step 7

Combine.

$\frac{7 b^{2} \sqrt{b}}{2 a^{2} \sqrt{3}}$

Step 8

Multiply $\frac{7 b^{2} \sqrt{b}}{2 a^{2} \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{7 b^{2} \sqrt{b}}{2 a^{2} \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{7 b^{2} \sqrt{b}}{2 a^{2} \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} \sqrt{3} \sqrt{3}}$

Step 9.2

Move $\sqrt{3}$ .

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} (\sqrt{3} \sqrt{3})}$

Step 9.3

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

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} ({\sqrt{3}}^{1} \sqrt{3})}$

Step 9.4

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

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 9.5

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

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} {\sqrt{3}}^{1 + 1}}$

Step 9.6

Add $1$ and $1$ .

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} {\sqrt{3}}^{2}}$

Step 9.7

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

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Step 9.7.1

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

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} {(3^{\frac{1}{2}})}^{2}}$

Step 9.7.2

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

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 9.7.3

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

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} \cdot 3^{\frac{2}{2}}}$

Step 9.7.4

Cancel the common factor of $2$ .

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Step 9.7.4.1

Cancel the common factor.

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} \cdot 3^{\frac{2}{2}}}$

Step 9.7.4.2

Rewrite the expression.

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} \cdot 3^{1}}$

Step 9.7.5

Evaluate the exponent.

$\frac{7 b^{2} \sqrt{b} \sqrt{3}}{2 a^{2} \cdot 3}$

Step 10

Combine using the product rule for radicals.

$\frac{7 b^{2} \sqrt{3 b}}{2 a^{2} \cdot 3}$

Step 11

Multiply $3$ by $2$ .

$\frac{7 b^{2} \sqrt{3 b}}{6 a^{2}}$