Algebra Examples

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

Step 1

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

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Step 1.1

Factor out $a^{4}$ .

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

Step 1.2

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

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

Step 1.3

Move $a$ .

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

Step 1.4

Move $6$ .

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

Step 1.5

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

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

Step 1.6

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Factor out $b^{2}$ .

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

Step 3.2

Move $a$ .

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

Step 3.3

Reorder $3$ and $b^{2}$ .

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

Step 3.4

Add parentheses.

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

Step 3.5

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $b$ .

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

Step 5.2

Multiply $b$ by $b$ .

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

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $6$ by $3$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Add $1$ and $1$ .

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

Step 7

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

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Step 7.1

Factor $9$ out of $18$ .

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

Step 7.2

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

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

Step 7.3

Move $2$ .

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

Step 7.4

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

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

Step 7.5

Add parentheses.

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

Step 8

Pull terms out from under the radical.

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

Step 9

Rewrite using the commutative property of multiplication.

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

Step 10

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

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Step 10.1

Move $a$ .

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

Step 10.2

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

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Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.3

Add $1$ and $2$ .

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