Simplify square root of 6c^9* square root of 2mc

$\sqrt{6 c^{9}} \cdot \sqrt{2 m c}$

Step 1

Rewrite $6 c^{9}$ as ${({(c^{2})}^{2})}^{2} \cdot (6 c)$ .

Factor out $c^{8}$ .

$\sqrt{6 (c^{8} c)} \cdot \sqrt{2 m c}$

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

$\sqrt{6 ({(c^{4})}^{2} c)} \cdot \sqrt{2 m c}$

Reorder $6$ and ${(c^{4})}^{2}$ .

$\sqrt{{(c^{4})}^{2} \cdot 6 c} \cdot \sqrt{2 m c}$

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

$\sqrt{{({(c^{2})}^{2})}^{2} \cdot 6 c} \cdot \sqrt{2 m c}$

Add parentheses.

$\sqrt{{({(c^{2})}^{2})}^{2} \cdot (6 c)} \cdot \sqrt{2 m c}$

Step 2

Pull terms out from under the radical.

${(c^{2})}^{2} \sqrt{6 c} \cdot \sqrt{2 m c}$

Step 3

Multiply the exponents in ${(c^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$c^{2 \cdot 2} \sqrt{6 c} \cdot \sqrt{2 m c}$

Multiply $2$ by $2$ .

$c^{4} \sqrt{6 c} \cdot \sqrt{2 m c}$

Step 4

Multiply $c^{4} \sqrt{6 c} \sqrt{2 m c}$ .

Combine using the product rule for radicals.

$c^{4} \sqrt{2 m c (6 c)}$

Multiply $6$ by $2$ .

$c^{4} \sqrt{12 m c \cdot c}$

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

$c^{4} \sqrt{12 m (c^{1} c)}$

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

$c^{4} \sqrt{12 m (c^{1} c^{1})}$

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

$c^{4} \sqrt{12 m c^{1 + 1}}$

Add $1$ and $1$ .

$c^{4} \sqrt{12 m c^{2}}$

Step 5

Rewrite $12 m c^{2}$ as ${(2 c)}^{2} \cdot (3 m)$ .

Factor $4$ out of $12$ .

$c^{4} \sqrt{4 (3) m c^{2}}$

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

$c^{4} \sqrt{2^{2} \cdot 3 m c^{2}}$

Move $m$ .

$c^{4} \sqrt{2^{2} \cdot 3 c^{2} m}$

Move $3$ .

$c^{4} \sqrt{2^{2} c^{2} \cdot 3 m}$

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

$c^{4} \sqrt{{(2 c)}^{2} \cdot 3 m}$

Add parentheses.

$c^{4} \sqrt{{(2 c)}^{2} \cdot (3 m)}$

Step 6

Pull terms out from under the radical.

$c^{4} (2 c \sqrt{3 m})$

Step 7

Rewrite using the commutative property of multiplication.

$2 c^{4} (c \sqrt{3 m})$

Step 8

Multiply $c^{4}$ by $c$ by adding the exponents.

Move $c$ .

$2 (c \cdot c^{4}) \sqrt{3 m}$

Multiply $c$ by $c^{4}$ .

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

$2 (c^{1} c^{4}) \sqrt{3 m}$

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

$2 c^{1 + 4} \sqrt{3 m}$

Add $1$ and $4$ .

$2 c^{5} \sqrt{3 m}$

Simplify square root of 6c^9* square root of 2mc

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