Simplify (9cd^5)(2c^3d)^2

$(9 c d^{5}) {(2 c^{3} d)}^{2}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 c^{3} d$ .

$9 c d^{5} ({(2 c^{3})}^{2} d^{2})$

Apply the product rule to $2 c^{3}$ .

$9 c d^{5} (2^{2} {(c^{3})}^{2} d^{2})$

Step 2

Multiply $d^{5}$ by $d^{2}$ by adding the exponents.

Move $d^{2}$ .

$9 c (d^{2} d^{5}) (2^{2} {(c^{3})}^{2})$

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

$9 c d^{2 + 5} (2^{2} {(c^{3})}^{2})$

Add $2$ and $5$ .

$9 c d^{7} (2^{2} {(c^{3})}^{2})$

Step 3

Raise $2$ to the power of $2$ .

$9 c d^{7} (4 {(c^{3})}^{2})$

Step 4

Multiply the exponents in ${(c^{3})}^{2}$ .

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

$9 c d^{7} (4 c^{3 \cdot 2})$

Multiply $3$ by $2$ .

$9 c d^{7} (4 c^{6})$

Step 5

Multiply $c$ by $c^{6}$ by adding the exponents.

Move $c^{6}$ .

$9 (c^{6} c) d^{7} \cdot 4$

Multiply $c^{6}$ by $c$ .

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

$9 (c^{6} c^{1}) d^{7} \cdot 4$

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

$9 c^{6 + 1} d^{7} \cdot 4$

Add $6$ and $1$ .

$9 c^{7} d^{7} \cdot 4$

Step 6

Multiply $4$ by $9$ .

$36 c^{7} d^{7}$