Evaluate (2cd^4)^2(cd)^5

${(2 c d^{4})}^{2} {(c d)}^{5}$

Step 1

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

Apply the product rule to $2 c d^{4}$ .

${(2 c)}^{2} {(d^{4})}^{2} {(c d)}^{5}$

Apply the product rule to $2 c$ .

$2^{2} c^{2} {(d^{4})}^{2} {(c d)}^{5}$

Step 2

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

$4 c^{2} {(d^{4})}^{2} {(c d)}^{5}$

Step 3

Multiply the exponents in ${(d^{4})}^{2}$ .

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

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

Multiply $4$ by $2$ .

$4 c^{2} d^{8} {(c d)}^{5}$

Step 4

Apply the product rule to $c d$ .

$4 c^{2} d^{8} (c^{5} d^{5})$

Step 5

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

Move $c^{5}$ .

$4 (c^{5} c^{2}) d^{8} d^{5}$

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

$4 c^{5 + 2} d^{8} d^{5}$

Add $5$ and $2$ .

$4 c^{7} d^{8} d^{5}$

Step 6

Multiply $d^{8}$ by $d^{5}$ by adding the exponents.

Move $d^{5}$ .

$4 c^{7} (d^{5} d^{8})$

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

$4 c^{7} d^{5 + 8}$

Add $5$ and $8$ .

$4 c^{7} d^{13}$