Algebra Examples

$\frac{\sqrt{40 a^{2} b^{5}}}{\sqrt{5 a^{5} b}}$

Step 1

Combine $\sqrt{40 a^{2} b^{5}}$ and $\sqrt{5 a^{5} b}$ into a single radical.

$\sqrt{\frac{40 a^{2} b^{5}}{5 a^{5} b}}$

Step 2

Reduce the expression $\frac{40 a^{2} b^{5}}{5 a^{5} b}$ by cancelling the common factors.

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Step 2.1

Factor $5$ out of $40 a^{2} b^{5}$ .

$\sqrt{\frac{5 (8 a^{2} b^{5})}{5 a^{5} b}}$

Step 2.2

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

$\sqrt{\frac{5 (8 a^{2} b^{5})}{5 (a^{5} b)}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{5 (8 a^{2} b^{5})}{5 (a^{5} b)}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{8 a^{2} b^{5}}{a^{5} b}}$

Step 3

Cancel the common factor of $a^{2}$ and $a^{5}$ .

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Step 3.1

Factor $a^{2}$ out of $8 a^{2} b^{5}$ .

$\sqrt{\frac{a^{2} (8 b^{5})}{a^{5} b}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\sqrt{\frac{a^{2} (8 b^{5})}{a^{2} (a^{3} b)}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{a^{2} (8 b^{5})}{a^{2} (a^{3} b)}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{8 b^{5}}{a^{3} b}}$

Step 4

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

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Step 4.1

Factor $b$ out of $8 b^{5}$ .

$\sqrt{\frac{b (8 b^{4})}{a^{3} b}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $b$ out of $a^{3} b$ .

$\sqrt{\frac{b (8 b^{4})}{b a^{3}}}$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{b (8 b^{4})}{b a^{3}}}$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{8 b^{4}}{a^{3}}}$

Step 5

Rewrite $\frac{8 b^{4}}{a^{3}}$ as ${(\frac{2 b^{2}}{a})}^{2} \frac{2}{a}$ .

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Step 5.1

Factor the perfect power ${(2 b^{2})}^{2}$ out of $8 b^{4}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Pull terms out from under the radical.

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

Step 7

Rewrite $\sqrt{\frac{2}{a}}$ as $\frac{\sqrt{2}}{\sqrt{a}}$ .

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

Step 8

Combine.

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

Step 9

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

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

Step 10

Combine and simplify the denominator.

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Step 10.1

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

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

Step 10.2

Move $\sqrt{a}$ .

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

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

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

Step 10.6

Add $1$ and $1$ .

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

Step 10.7

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

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Step 10.7.1

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

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

Step 10.7.2

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

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

Step 10.7.3

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

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

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

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

Step 10.7.4.2

Rewrite the expression.

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

Step 10.7.5

Simplify.

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

Step 11

Multiply $a$ by $a$ .

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

Step 12

Combine using the product rule for radicals.

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

Step 13

Move $2$ to the left of $a$ .

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