Algebra Examples

$\sqrt{a^{3} b^{5}} \cdot 5 \sqrt{4 a b}$

Step 1

Rewrite using the commutative property of multiplication.

$5 \sqrt{a^{3} b^{5}} \sqrt{4 a b}$

Step 2

Rewrite $a^{3} b^{5}$ as ${(a b^{2})}^{2} (a b)$ .

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Step 2.1

Factor out $a^{2}$ .

$5 \sqrt{a^{2} a b^{5}} \sqrt{4 a b}$

Step 2.2

Factor out $b^{4}$ .

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

Step 2.3

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

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

Step 2.4

Move $a$ .

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

Step 2.5

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

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

Step 2.6

Add parentheses.

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

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Step 4.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

Multiply $5 a b^{2} \sqrt{a b} (2 \sqrt{a b})$ .

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Step 6.1

Multiply $2$ by $5$ .

$10 a b^{2} \sqrt{a b} \sqrt{a b}$

Step 6.2

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

$10 a b^{2} ({\sqrt{a b}}^{1} \sqrt{a b})$

Step 6.3

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

$10 a b^{2} ({\sqrt{a b}}^{1} {\sqrt{a b}}^{1})$

Step 6.4

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

$10 a b^{2} {\sqrt{a b}}^{1 + 1}$

Step 6.5

Add $1$ and $1$ .

$10 a b^{2} {\sqrt{a b}}^{2}$

Step 7

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

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Step 7.1

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

$10 a b^{2} {({(a b)}^{\frac{1}{2}})}^{2}$

Step 7.2

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

$10 a b^{2} {(a b)}^{\frac{1}{2} \cdot 2}$

Step 7.3

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

$10 a b^{2} {(a b)}^{\frac{2}{2}}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$10 a b^{2} {(a b)}^{\frac{2}{2}}$

Step 7.4.2

Rewrite the expression.

$10 a b^{2} {(a b)}^{1}$

Step 7.5

Simplify.

$10 a b^{2} (a b)$

Step 8

Multiply $a$ by $a$ by adding the exponents.

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Step 8.1

Move $a$ .

$10 (a \cdot a) b^{2} b$

Step 8.2

Multiply $a$ by $a$ .

$10 a^{2} b^{2} b$

Step 9

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

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Step 9.1

Move $b$ .

$10 a^{2} (b \cdot b^{2})$

Step 9.2

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

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Step 9.2.1

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

$10 a^{2} (b^{1} b^{2})$

Step 9.2.2

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

$10 a^{2} b^{1 + 2}$

Step 9.3

Add $1$ and $2$ .

$10 a^{2} b^{3}$