Simplify square root of 4c^3d^3* square root of 8c^3d

$\sqrt{4 c^{3} d^{3}} \cdot \sqrt{8 c^{3} d}$

Step 1

Rewrite $4 c^{3} d^{3}$ as ${(2 c d)}^{2} (c d)$ .

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

$\sqrt{2^{2} c^{3} d^{3}} \cdot \sqrt{8 c^{3} d}$

Factor out $c^{2}$ .

$\sqrt{2^{2} (c^{2} c) d^{3}} \cdot \sqrt{8 c^{3} d}$

Factor out $d^{2}$ .

$\sqrt{2^{2} (c^{2} c) (d^{2} d)} \cdot \sqrt{8 c^{3} d}$

Move $c$ .

$\sqrt{2^{2} (c^{2}) d^{2} c d} \cdot \sqrt{8 c^{3} d}$

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

$\sqrt{{(2 c d)}^{2} c d} \cdot \sqrt{8 c^{3} d}$

Add parentheses.

$\sqrt{{(2 c d)}^{2} (c d)} \cdot \sqrt{8 c^{3} d}$

Step 2

Pull terms out from under the radical.

$2 c d \sqrt{c d} \cdot \sqrt{8 c^{3} d}$

Step 3

Rewrite $8 c^{3} d$ as ${(2 c)}^{2} \cdot (2 c d)$ .

Factor $4$ out of $8$ .

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

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

$2 c d \sqrt{c d} \cdot \sqrt{2^{2} \cdot 2 c^{3} d}$

Factor out $c^{2}$ .

$2 c d \sqrt{c d} \cdot \sqrt{2^{2} \cdot 2 (c^{2} c) d}$

Move $2$ .

$2 c d \sqrt{c d} \cdot \sqrt{2^{2} c^{2} \cdot 2 c d}$

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

$2 c d \sqrt{c d} \cdot \sqrt{{(2 c)}^{2} \cdot 2 c d}$

Add parentheses.

$2 c d \sqrt{c d} \cdot \sqrt{{(2 c)}^{2} \cdot 2 (c d)}$

Add parentheses.

$2 c d \sqrt{c d} \cdot \sqrt{{(2 c)}^{2} \cdot (2 c d)}$

Step 4

Pull terms out from under the radical.

$2 c d \sqrt{c d} \cdot (2 c \sqrt{2 c d})$

Step 5

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

$2 (c \cdot c) d \sqrt{c d} \cdot (2 \sqrt{2 c d})$

Multiply $c$ by $c$ .

$2 c^{2} d \sqrt{c d} \cdot (2 \sqrt{2 c d})$

Step 6

Multiply $2 c^{2} d \sqrt{c d} (2 \sqrt{2 c d})$ .

Multiply $2$ by $2$ .

$4 c^{2} d \sqrt{c d} \sqrt{2 c d}$

Combine using the product rule for radicals.

$4 c^{2} d \sqrt{2 c d (c d)}$

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

$4 c^{2} d \sqrt{2 (c^{1} c) d \cdot d}$

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

$4 c^{2} d \sqrt{2 (c^{1} c^{1}) d \cdot d}$

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

$4 c^{2} d \sqrt{2 c^{1 + 1} d \cdot d}$

Add $1$ and $1$ .

$4 c^{2} d \sqrt{2 c^{2} d \cdot d}$

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

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

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

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

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

$4 c^{2} d \sqrt{2 c^{2} d^{1 + 1}}$

Add $1$ and $1$ .

$4 c^{2} d \sqrt{2 c^{2} d^{2}}$

Step 7

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

Move $2$ .

$4 c^{2} d \sqrt{c^{2} d^{2} \cdot 2}$

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

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

Step 8

Pull terms out from under the radical.

$4 c^{2} d (c d \sqrt{2})$

Step 9

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

Move $c$ .

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

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

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

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

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

$4 c^{1 + 2} d (d \sqrt{2})$

Add $1$ and $2$ .

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

Step 10

Multiply $d$ by $d$ by adding the exponents.

Move $d$ .

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

Multiply $d$ by $d$ .

$4 c^{3} d^{2} \sqrt{2}$

Simplify square root of 4c^3d^3* square root of 8c^3d

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