Simplify 6 square root of 8c^3d^5*4 square root of 2cd^3

$6 \sqrt{8 c^{3} d^{5}} \cdot 4 \sqrt{2 c d^{3}}$

Step 1

Rewrite $8 c^{3} d^{5}$ as ${(2 c d^{2})}^{2} \cdot (2 c d)$ .

Factor $4$ out of $8$ .

$6 \sqrt{4 (2) c^{3} d^{5}} \cdot (4 \sqrt{2 c d^{3}})$

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

$6 \sqrt{2^{2} \cdot 2 c^{3} d^{5}} \cdot (4 \sqrt{2 c d^{3}})$

Factor out $c^{2}$ .

$6 \sqrt{2^{2} \cdot 2 (c^{2} c) d^{5}} \cdot (4 \sqrt{2 c d^{3}})$

Factor out $d^{4}$ .

$6 \sqrt{2^{2} \cdot 2 (c^{2} c) (d^{4} d)} \cdot (4 \sqrt{2 c d^{3}})$

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

$6 \sqrt{2^{2} \cdot 2 (c^{2} c) ({(d^{2})}^{2} d)} \cdot (4 \sqrt{2 c d^{3}})$

Move $c$ .

$6 \sqrt{2^{2} \cdot 2 (c^{2}) {(d^{2})}^{2} c d} \cdot (4 \sqrt{2 c d^{3}})$

Move $2$ .

$6 \sqrt{2^{2} (c^{2}) {(d^{2})}^{2} \cdot 2 c d} \cdot (4 \sqrt{2 c d^{3}})$

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

$6 \sqrt{{(2 c d^{2})}^{2} \cdot 2 c d} \cdot (4 \sqrt{2 c d^{3}})$

Add parentheses.

$6 \sqrt{{(2 c d^{2})}^{2} \cdot 2 (c d)} \cdot (4 \sqrt{2 c d^{3}})$

Add parentheses.

$6 \sqrt{{(2 c d^{2})}^{2} \cdot (2 c d)} \cdot (4 \sqrt{2 c d^{3}})$

Step 2

Pull terms out from under the radical.

$6 (2 c d^{2} \sqrt{2 c d}) \cdot (4 \sqrt{2 c d^{3}})$

Step 3

Multiply $2$ by $6$ .

$12 (c d^{2} \sqrt{2 c d}) \cdot (4 \sqrt{2 c d^{3}})$

Step 4

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

Factor out $d^{2}$ .

$12 c d^{2} \sqrt{2 c d} \cdot (4 \sqrt{2 c (d^{2} d)})$

Move $c$ .

$12 c d^{2} \sqrt{2 c d} \cdot (4 \sqrt{2 d^{2} c d})$

Reorder $2$ and $d^{2}$ .

$12 c d^{2} \sqrt{2 c d} \cdot (4 \sqrt{d^{2} \cdot 2 c d})$

Add parentheses.

$12 c d^{2} \sqrt{2 c d} \cdot (4 \sqrt{d^{2} \cdot 2 (c d)})$

Add parentheses.

$12 c d^{2} \sqrt{2 c d} \cdot (4 \sqrt{d^{2} \cdot (2 c d)})$

Step 5

Pull terms out from under the radical.

$12 c d^{2} \sqrt{2 c d} \cdot (4 (d \sqrt{2 c d}))$

Step 6

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

Move $d$ .

$12 c (d \cdot d^{2}) \sqrt{2 c d} \cdot (4 (\sqrt{2 c d}))$

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

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

$12 c (d^{1} d^{2}) \sqrt{2 c d} \cdot (4 (\sqrt{2 c d}))$

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

$12 c d^{1 + 2} \sqrt{2 c d} \cdot (4 (\sqrt{2 c d}))$

Add $1$ and $2$ .

$12 c d^{3} \sqrt{2 c d} \cdot (4 (\sqrt{2 c d}))$

Step 7

Multiply $12 c d^{3} \sqrt{2 c d} (4 \sqrt{2 c d})$ .

Multiply $4$ by $12$ .

$48 c d^{3} \sqrt{2 c d} \sqrt{2 c d}$

Raise $\sqrt{2 c d}$ to the power of $1$ .

$48 c d^{3} ({\sqrt{2 c d}}^{1} \sqrt{2 c d})$

Raise $\sqrt{2 c d}$ to the power of $1$ .

$48 c d^{3} ({\sqrt{2 c d}}^{1} {\sqrt{2 c d}}^{1})$

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

$48 c d^{3} {\sqrt{2 c d}}^{1 + 1}$

Add $1$ and $1$ .

$48 c d^{3} {\sqrt{2 c d}}^{2}$

Step 8

Rewrite ${\sqrt{2 c d}}^{2}$ as $2 c d$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 c d}$ as ${(2 c d)}^{\frac{1}{2}}$ .

$48 c d^{3} {({(2 c d)}^{\frac{1}{2}})}^{2}$

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

$48 c d^{3} {(2 c d)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$48 c d^{3} {(2 c d)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$48 c d^{3} {(2 c d)}^{\frac{2}{2}}$

Rewrite the expression.

$48 c d^{3} {(2 c d)}^{1}$

Simplify.

$48 c d^{3} (2 c d)$

Step 9

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

Move $c$ .

$48 (c \cdot c) d^{3} (2 d)$

Multiply $c$ by $c$ .

$48 c^{2} d^{3} (2 d)$

Step 10

Multiply $d^{3}$ by $d$ by adding the exponents.

Move $d$ .

$48 c^{2} (d \cdot d^{3}) \cdot 2$

Multiply $d$ by $d^{3}$ .

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

$48 c^{2} (d^{1} d^{3}) \cdot 2$

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

$48 c^{2} d^{1 + 3} \cdot 2$

Add $1$ and $3$ .

$48 c^{2} d^{4} \cdot 2$

Step 11

Multiply $2$ by $48$ .

$96 c^{2} d^{4}$

Simplify 6 square root of 8c^3d^5*4 square root of 2cd^3

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