Algebra Examples

$\sqrt{\frac{9 a b}{4 a b^{4}}}$

Step 1

Reduce the expression $\frac{9 a b}{4 a b^{4}}$ by cancelling the common factors.

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Step 1.1

Cancel the common factor.

$\sqrt{\frac{9 a b}{4 a b^{4}}}$

Step 1.2

Rewrite the expression.

$\sqrt{\frac{9 b}{4 b^{4}}}$

Step 2

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

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Step 2.1

Factor $b$ out of $9 b$ .

$\sqrt{\frac{b \cdot 9}{4 b^{4}}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

$\sqrt{\frac{b \cdot 9}{b (4 b^{3})}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{b \cdot 9}{b (4 b^{3})}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{9}{4 b^{3}}}$

Step 3

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

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Step 3.1

Factor the perfect power $3^{2}$ out of $9$ .

$\sqrt{\frac{3^{2} \cdot 1}{4 b^{3}}}$

Step 3.2

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

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

Step 3.3

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

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

Step 4

Pull terms out from under the radical.

$\frac{3}{2 b} \sqrt{\frac{1}{b}}$

Step 5

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

$\frac{3}{2 b} \cdot \frac{\sqrt{1}}{\sqrt{b}}$

Step 6

Any root of $1$ is $1$ .

$\frac{3}{2 b} \cdot \frac{1}{\sqrt{b}}$

Step 7

Multiply $\frac{1}{\sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

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

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{1}{\sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{3}{2 b} \cdot \frac{\sqrt{b}}{\sqrt{b} \sqrt{b}}$

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Simplify.

$\frac{3}{2 b} \cdot \frac{\sqrt{b}}{b}$

Step 9

Multiply $\frac{3}{2 b} \cdot \frac{\sqrt{b}}{b}$ .

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Step 9.1

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

$\frac{3 \sqrt{b}}{2 b \cdot b}$

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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