Simplify square root of 12b^6 square root of 30b^5

$\sqrt{12 b^{6}} \sqrt{30 b^{5}}$

Step 1

Rewrite $12 b^{6}$ as ${(2 b^{3})}^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\sqrt{4 (3) b^{6}} \sqrt{30 b^{5}}$

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

$\sqrt{2^{2} \cdot 3 b^{6}} \sqrt{30 b^{5}}$

Rewrite $b^{6}$ as ${(b^{3})}^{2}$ .

$\sqrt{2^{2} \cdot 3 {(b^{3})}^{2}} \sqrt{30 b^{5}}$

Move $3$ .

$\sqrt{2^{2} {(b^{3})}^{2} \cdot 3} \sqrt{30 b^{5}}$

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

$\sqrt{{(2 b^{3})}^{2} \cdot 3} \sqrt{30 b^{5}}$

Step 2

Pull terms out from under the radical.

$2 b^{3} \sqrt{3} \sqrt{30 b^{5}}$

Step 3

Rewrite $30 b^{5}$ as ${(b^{2})}^{2} \cdot (30 b)$ .

Factor out $b^{4}$ .

$2 b^{3} \sqrt{3} \sqrt{30 (b^{4} b)}$

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

$2 b^{3} \sqrt{3} \sqrt{30 ({(b^{2})}^{2} b)}$

Reorder $30$ and ${(b^{2})}^{2}$ .

$2 b^{3} \sqrt{3} \sqrt{{(b^{2})}^{2} \cdot 30 b}$

Add parentheses.

$2 b^{3} \sqrt{3} \sqrt{{(b^{2})}^{2} \cdot (30 b)}$

Step 4

Pull terms out from under the radical.

$2 b^{3} \sqrt{3} (b^{2} \sqrt{30 b})$

Step 5

Multiply $b^{3}$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

$2 (b^{2} b^{3}) \sqrt{3} \sqrt{30 b}$

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

$2 b^{2 + 3} \sqrt{3} \sqrt{30 b}$

Add $2$ and $3$ .

$2 b^{5} \sqrt{3} \sqrt{30 b}$

Step 6

Multiply $2 b^{5} \sqrt{3} \sqrt{30 b}$ .

Combine using the product rule for radicals.

$2 b^{5} \sqrt{30 b \cdot 3}$

Multiply $3$ by $30$ .

$2 b^{5} \sqrt{90 b}$

Step 7

Rewrite $90 b$ as $3^{2} \cdot (10 b)$ .

Factor $9$ out of $90$ .

$2 b^{5} \sqrt{9 (10) b}$

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

$2 b^{5} \sqrt{3^{2} \cdot 10 b}$

Add parentheses.

$2 b^{5} \sqrt{3^{2} \cdot (10 b)}$

Step 8

Pull terms out from under the radical.

$2 b^{5} (3 \sqrt{10 b})$

Step 9

Multiply $3$ by $2$ .

$6 b^{5} \sqrt{10 b}$

Simplify square root of 12b^6 square root of 30b^5

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