Algebra Examples

$\frac{\sqrt{a^{9}}}{\sqrt{b^{5}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $a^{9}$ as ${(a^{4})}^{2} a$ .

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Step 1.1.1

Factor out $a^{8}$ .

$\frac{\sqrt{a^{8} a}}{\sqrt{b^{5}}}$

Step 1.1.2

Rewrite $a^{8}$ as ${(a^{4})}^{2}$ .

$\frac{\sqrt{{(a^{4})}^{2} a}}{\sqrt{b^{5}}}$

Step 1.2

Pull terms out from under the radical.

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $b^{5}$ as ${(b^{2})}^{2} b$ .

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Step 2.1.1

Factor out $b^{4}$ .

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

Step 2.1.2

Rewrite $b^{4}$ as ${(b^{2})}^{2}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

Multiply $\frac{a^{4} \sqrt{a}}{b^{2} \sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

$\frac{a^{4} \sqrt{a}}{b^{2} \sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{a^{4} \sqrt{a}}{b^{2} \sqrt{b}}$ by $\frac{\sqrt{b}}{\sqrt{b}}$ .

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

Step 4.2

Move $\sqrt{b}$ .

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

Step 4.3

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

$\frac{a^{4} \sqrt{a} \sqrt{b}}{b^{2} ({\sqrt{b}}^{1} \sqrt{b})}$

Step 4.4

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

$\frac{a^{4} \sqrt{a} \sqrt{b}}{b^{2} ({\sqrt{b}}^{1} {\sqrt{b}}^{1})}$

Step 4.5

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

$\frac{a^{4} \sqrt{a} \sqrt{b}}{b^{2} {\sqrt{b}}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

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

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Step 4.7.1

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

$\frac{a^{4} \sqrt{a} \sqrt{b}}{b^{2} {(b^{\frac{1}{2}})}^{2}}$

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

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

Step 4.7.4.2

Rewrite the expression.

$\frac{a^{4} \sqrt{a} \sqrt{b}}{b^{2} b^{1}}$

Step 4.7.5

Simplify.

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

Step 5

Multiply $b^{2}$ by $b$ by adding the exponents.

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Step 5.1

Multiply $b^{2}$ by $b$ .

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Step 5.1.1

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

$\frac{a^{4} \sqrt{a} \sqrt{b}}{b^{2} b^{1}}$

Step 5.1.2

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

$\frac{a^{4} \sqrt{a} \sqrt{b}}{b^{2 + 1}}$

Step 5.2

Add $2$ and $1$ .

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

Step 6

Combine using the product rule for radicals.

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