Basic Math Examples

$a^{2} \sqrt{16 a b^{9}} - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $16 a b^{9}$ as ${(4 b^{4})}^{2} (a b)$ .

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Step 1.1.1

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

$a^{2} \sqrt{4^{2} a b^{9}} - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.1.2

Factor out $b^{8}$ .

$a^{2} \sqrt{4^{2} a (b^{8} b)} - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.1.3

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

$a^{2} \sqrt{4^{2} a ({(b^{4})}^{2} b)} - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.1.4

Move $a$ .

$a^{2} \sqrt{4^{2} {(b^{4})}^{2} a b} - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.1.5

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

$a^{2} \sqrt{{(4 b^{4})}^{2} a b} - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.1.6

Add parentheses.

$a^{2} \sqrt{{(4 b^{4})}^{2} (a b)} - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.2

Pull terms out from under the radical.

$a^{2} (4 b^{4} \sqrt{a b}) - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.3

Rewrite using the commutative property of multiplication.

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{4 a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.4

Rewrite $4 a^{5} b^{9}$ as ${(2 a^{2} b^{4})}^{2} (a b)$ .

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Step 1.4.1

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

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{2^{2} a^{5} b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.4.2

Factor out $a^{4}$ .

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{2^{2} (a^{4} a) b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.4.3

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

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{2^{2} ({(a^{2})}^{2} a) b^{9}} + \sqrt{64 a^{5} b^{9}}$

Step 1.4.4

Factor out $b^{8}$ .

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{2^{2} ({(a^{2})}^{2} a) (b^{8} b)} + \sqrt{64 a^{5} b^{9}}$

Step 1.4.5

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

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{2^{2} ({(a^{2})}^{2} a) ({(b^{4})}^{2} b)} + \sqrt{64 a^{5} b^{9}}$

Step 1.4.6

Move $a$ .

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{2^{2} ({(a^{2})}^{2}) {(b^{4})}^{2} a b} + \sqrt{64 a^{5} b^{9}}$

Step 1.4.7

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

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{{(2 a^{2} b^{4})}^{2} a b} + \sqrt{64 a^{5} b^{9}}$

Step 1.4.8

Add parentheses.

$4 a^{2} (b^{4} \sqrt{a b}) - \sqrt{{(2 a^{2} b^{4})}^{2} (a b)} + \sqrt{64 a^{5} b^{9}}$

Step 1.5

Pull terms out from under the radical.

$4 a^{2} (b^{4} \sqrt{a b}) - (2 a^{2} b^{4} \sqrt{a b}) + \sqrt{64 a^{5} b^{9}}$

Step 1.6

Multiply $2$ by $- 1$ .

$4 a^{2} (b^{4} \sqrt{a b}) - 2 (a^{2} b^{4} \sqrt{a b}) + \sqrt{64 a^{5} b^{9}}$

Step 1.7

Rewrite $64 a^{5} b^{9}$ as ${(8 a^{2} b^{4})}^{2} (a b)$ .

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Step 1.7.1

Rewrite $64$ as $8^{2}$ .

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + \sqrt{8^{2} a^{5} b^{9}}$

Step 1.7.2

Factor out $a^{4}$ .

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + \sqrt{8^{2} (a^{4} a) b^{9}}$

Step 1.7.3

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

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + \sqrt{8^{2} ({(a^{2})}^{2} a) b^{9}}$

Step 1.7.4

Factor out $b^{8}$ .

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + \sqrt{8^{2} ({(a^{2})}^{2} a) (b^{8} b)}$

Step 1.7.5

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

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + \sqrt{8^{2} ({(a^{2})}^{2} a) ({(b^{4})}^{2} b)}$

Step 1.7.6

Move $a$ .

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + \sqrt{8^{2} ({(a^{2})}^{2}) {(b^{4})}^{2} a b}$

Step 1.7.7

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

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + \sqrt{{(8 a^{2} b^{4})}^{2} a b}$

Step 1.7.8

Add parentheses.

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + \sqrt{{(8 a^{2} b^{4})}^{2} (a b)}$

Step 1.8

Pull terms out from under the radical.

$4 a^{2} (b^{4} \sqrt{a b}) - 2 a^{2} b^{4} \sqrt{a b} + 8 a^{2} b^{4} \sqrt{a b}$

Step 2

Subtract $2 a^{2} b^{4} \sqrt{a b}$ from $4 a^{2} b^{4} \sqrt{a b}$ .

$2 a^{2} b^{4} \sqrt{a b} + 8 a^{2} b^{4} \sqrt{a b}$

Step 3

Add $2 a^{2} b^{4} \sqrt{a b}$ and $8 a^{2} b^{4} \sqrt{a b}$ .

$10 a^{2} b^{4} \sqrt{a b}$