Basic Math Examples

$\sqrt{28 a^{11} b} \cdot \sqrt{8 a^{12} b^{10}}$

Step 1

Rewrite $28 a^{11} b$ as ${(2 a^{5})}^{2} \cdot (7 a b)$ .

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Step 1.1

Factor $4$ out of $28$ .

$\sqrt{4 (7) a^{11} b} \cdot \sqrt{8 a^{12} b^{10}}$

Step 1.2

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

$\sqrt{2^{2} \cdot 7 a^{11} b} \cdot \sqrt{8 a^{12} b^{10}}$

Step 1.3

Factor out $a^{10}$ .

$\sqrt{2^{2} \cdot 7 (a^{10} a) b} \cdot \sqrt{8 a^{12} b^{10}}$

Step 1.4

Rewrite $a^{10}$ as ${(a^{5})}^{2}$ .

$\sqrt{2^{2} \cdot 7 ({(a^{5})}^{2} a) b} \cdot \sqrt{8 a^{12} b^{10}}$

Step 1.5

Move $7$ .

$\sqrt{2^{2} {(a^{5})}^{2} \cdot 7 a b} \cdot \sqrt{8 a^{12} b^{10}}$

Step 1.6

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

$\sqrt{{(2 a^{5})}^{2} \cdot 7 a b} \cdot \sqrt{8 a^{12} b^{10}}$

Step 1.7

Add parentheses.

$\sqrt{{(2 a^{5})}^{2} \cdot 7 (a b)} \cdot \sqrt{8 a^{12} b^{10}}$

Step 1.8

Add parentheses.

$\sqrt{{(2 a^{5})}^{2} \cdot (7 a b)} \cdot \sqrt{8 a^{12} b^{10}}$

Step 2

Pull terms out from under the radical.

$2 a^{5} \sqrt{7 a b} \cdot \sqrt{8 a^{12} b^{10}}$

Step 3

Rewrite $8 a^{12} b^{10}$ as ${(2 a^{6} b^{5})}^{2} \cdot 2$ .

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Step 3.1

Factor $4$ out of $8$ .

$2 a^{5} \sqrt{7 a b} \cdot \sqrt{4 (2) a^{12} b^{10}}$

Step 3.2

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

$2 a^{5} \sqrt{7 a b} \cdot \sqrt{2^{2} \cdot 2 a^{12} b^{10}}$

Step 3.3

Rewrite $a^{12}$ as ${(a^{6})}^{2}$ .

$2 a^{5} \sqrt{7 a b} \cdot \sqrt{2^{2} \cdot 2 {(a^{6})}^{2} b^{10}}$

Step 3.4

Rewrite $b^{10}$ as ${(b^{5})}^{2}$ .

$2 a^{5} \sqrt{7 a b} \cdot \sqrt{2^{2} \cdot 2 {(a^{6})}^{2} {(b^{5})}^{2}}$

Step 3.5

Move $2$ .

$2 a^{5} \sqrt{7 a b} \cdot \sqrt{2^{2} {(a^{6})}^{2} {(b^{5})}^{2} \cdot 2}$

Step 3.6

Rewrite $2^{2} {(a^{6})}^{2} {(b^{5})}^{2}$ as ${(2 a^{6} b^{5})}^{2}$ .

$2 a^{5} \sqrt{7 a b} \cdot \sqrt{{(2 a^{6} b^{5})}^{2} \cdot 2}$

Step 4

Pull terms out from under the radical.

$2 a^{5} \sqrt{7 a b} \cdot (2 a^{6} b^{5} \sqrt{2})$

Step 5

Multiply $a^{5}$ by $a^{6}$ by adding the exponents.

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Step 5.1

Move $a^{6}$ .

$2 (a^{6} a^{5}) \sqrt{7 a b} \cdot (2 b^{5} \sqrt{2})$

Step 5.2

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

$2 a^{6 + 5} \sqrt{7 a b} \cdot (2 b^{5} \sqrt{2})$

Step 5.3

Add $6$ and $5$ .

$2 a^{11} \sqrt{7 a b} \cdot (2 b^{5} \sqrt{2})$

Step 6

Multiply $2 a^{11} \sqrt{7 a b} (2 b^{5} \sqrt{2})$ .

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Step 6.1

Multiply $2$ by $2$ .

$4 a^{11} \sqrt{7 a b} (b^{5} \sqrt{2})$

Step 6.2

Combine using the product rule for radicals.

$4 a^{11} (b^{5} \sqrt{7 a b \cdot 2})$

Step 6.3

Multiply $2$ by $7$ .

$4 a^{11} (b^{5} \sqrt{14 a b})$

Step 7

Reorder factors in $4 a^{11} (b^{5} \sqrt{14 a b})$ .

$4 a^{11} b^{5} \sqrt{14 a b}$