Simplify (2de^5)^2(de)^4

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

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

Apply the product rule to $2 d e^{5}$ .

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

Apply the product rule to $2 d$ .

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

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

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

Multiply the exponents in ${(e^{5})}^{2}$ .

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

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

Multiply $5$ by $2$ .

$4 d^{2} e^{10} {(d e)}^{4}$

Apply the product rule to $d e$ .

$4 d^{2} e^{10} (d^{4} e^{4})$

Multiply $d^{2}$ by $d^{4}$ by adding the exponents.

Move $d^{4}$ .

$4 (d^{4} d^{2}) e^{10} e^{4}$

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

$4 d^{4 + 2} e^{10} e^{4}$

Add $4$ and $2$ .

$4 d^{6} e^{10} e^{4}$

Multiply $e^{10}$ by $e^{4}$ by adding the exponents.

Move $e^{4}$ .

$4 d^{6} (e^{4} e^{10})$

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

$4 d^{6} e^{4 + 10}$

Add $4$ and $10$ .

$4 d^{6} e^{14}$