Basic Math Examples

$\sqrt{6 a b} \cdot \sqrt{12 a^{5} b}$

Step 1

Rewrite $12 a^{5} b$ as ${(2 a^{2})}^{2} \cdot (3 a b)$ .

Tap for more steps...

Step 1.1

Factor $4$ out of $12$ .

$\sqrt{6 a b} \cdot \sqrt{4 (3) a^{5} b}$

Step 1.2

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

$\sqrt{6 a b} \cdot \sqrt{2^{2} \cdot 3 a^{5} b}$

Step 1.3

Factor out $a^{4}$ .

$\sqrt{6 a b} \cdot \sqrt{2^{2} \cdot 3 (a^{4} a) b}$

Step 1.4

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

$\sqrt{6 a b} \cdot \sqrt{2^{2} \cdot 3 ({(a^{2})}^{2} a) b}$

Step 1.5

Move $3$ .

$\sqrt{6 a b} \cdot \sqrt{2^{2} {(a^{2})}^{2} \cdot 3 a b}$

Step 1.6

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

$\sqrt{6 a b} \cdot \sqrt{{(2 a^{2})}^{2} \cdot 3 a b}$

Step 1.7

Add parentheses.

$\sqrt{6 a b} \cdot \sqrt{{(2 a^{2})}^{2} \cdot 3 (a b)}$

Step 1.8

Add parentheses.

$\sqrt{6 a b} \cdot \sqrt{{(2 a^{2})}^{2} \cdot (3 a b)}$

Step 2

Pull terms out from under the radical.

$\sqrt{6 a b} \cdot (2 a^{2} \sqrt{3 a b})$

Step 3

Rewrite using the commutative property of multiplication.

$2 \sqrt{6 a b} (a^{2} \sqrt{3 a b})$

Step 4

Multiply $2 \sqrt{6 a b} (a^{2} \sqrt{3 a b})$ .

Tap for more steps...

Step 4.1

Combine using the product rule for radicals.

$2 (a^{2} \sqrt{6 a b (3 a b)})$

Step 4.2

Multiply $3$ by $6$ .

$2 (a^{2} \sqrt{18 a b (a b)})$

Step 4.3

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

$2 (a^{2} \sqrt{18 (a^{1} a) b \cdot b})$

Step 4.4

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

$2 (a^{2} \sqrt{18 (a^{1} a^{1}) b \cdot b})$

Step 4.5

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

$2 (a^{2} \sqrt{18 a^{1 + 1} b \cdot b})$

Step 4.6

Add $1$ and $1$ .

$2 (a^{2} \sqrt{18 a^{2} b \cdot b})$

Step 4.7

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

$2 (a^{2} \sqrt{18 a^{2} (b^{1} b)})$

Step 4.8

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

$2 (a^{2} \sqrt{18 a^{2} (b^{1} b^{1})})$

Step 4.9

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

$2 (a^{2} \sqrt{18 a^{2} b^{1 + 1}})$

Step 4.10

Add $1$ and $1$ .

$2 (a^{2} \sqrt{18 a^{2} b^{2}})$

Step 5

Rewrite $18 a^{2} b^{2}$ as ${(3 a b)}^{2} \cdot 2$ .

Tap for more steps...

Step 5.1

Factor $9$ out of $18$ .

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

Step 5.2

Rewrite $9$ as $3^{2}$ .

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

Step 5.3

Move $2$ .

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

Step 5.4

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

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

Step 6

Pull terms out from under the radical.

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

Step 7

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

Tap for more steps...

Step 7.1

Move $a$ .

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

Step 7.2

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

Tap for more steps...

Step 7.2.1

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

$2 (a^{1} a^{2} (3 b \sqrt{2}))$

Step 7.2.2

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

$2 (a^{1 + 2} (3 b \sqrt{2}))$

Step 7.3

Add $1$ and $2$ .

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

Step 8

Rewrite using the commutative property of multiplication.

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

Step 9

Multiply $3$ by $2$ .

$6 \sqrt{2} a^{3} b$