Basic Math Examples

$\sqrt{16 a^{3} b^{5}} + 5 a b \sqrt{a b^{3}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $16 a^{3} b^{5}$ as ${(4 a b^{2})}^{2} (a b)$ .

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Step 1.1.1

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} a^{3} b^{5}} + 5 a b \sqrt{a b^{3}}$

Step 1.1.2

Factor out $a^{2}$ .

$\sqrt{4^{2} (a^{2} a) b^{5}} + 5 a b \sqrt{a b^{3}}$

Step 1.1.3

Factor out $b^{4}$ .

$\sqrt{4^{2} (a^{2} a) (b^{4} b)} + 5 a b \sqrt{a b^{3}}$

Step 1.1.4

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

$\sqrt{4^{2} (a^{2} a) ({(b^{2})}^{2} b)} + 5 a b \sqrt{a b^{3}}$

Step 1.1.5

Move $a$ .

$\sqrt{4^{2} (a^{2}) {(b^{2})}^{2} a b} + 5 a b \sqrt{a b^{3}}$

Step 1.1.6

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

$\sqrt{{(4 a b^{2})}^{2} a b} + 5 a b \sqrt{a b^{3}}$

Step 1.1.7

Add parentheses.

$\sqrt{{(4 a b^{2})}^{2} (a b)} + 5 a b \sqrt{a b^{3}}$

Step 1.2

Pull terms out from under the radical.

$4 a b^{2} \sqrt{a b} + 5 a b \sqrt{a b^{3}}$

Step 1.3

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

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Step 1.3.1

Factor out $b^{2}$ .

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

Step 1.3.2

Reorder $a$ and $b^{2}$ .

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

Step 1.3.3

Add parentheses.

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

Step 1.4

Pull terms out from under the radical.

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

Step 1.5

Multiply $b$ by $b$ by adding the exponents.

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Step 1.5.1

Move $b$ .

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

Step 1.5.2

Multiply $b$ by $b$ .

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

Step 2

Add $4 a b^{2} \sqrt{a b}$ and $5 a b^{2} \sqrt{a b}$ .

$9 a b^{2} \sqrt{a b}$