Simplify 17(a^(2n+1))^4*2a^(7n-12)

$17 {(a^{2 n + 1})}^{4} \cdot 2 a^{7 n - 12}$

Rewrite using the commutative property of multiplication.

$17 \cdot 2 {(a^{2 n + 1})}^{4} a^{7 n - 12}$

Multiply $17$ by $2$ .

$34 {(a^{2 n + 1})}^{4} a^{7 n - 12}$

Multiply the exponents in ${(a^{2 n + 1})}^{4}$ .

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

$34 a^{(2 n + 1) \cdot 4} a^{7 n - 12}$

Apply the distributive property.

$34 a^{2 n \cdot 4 + 1 \cdot 4} a^{7 n - 12}$

Multiply $4$ by $2$ .

$34 a^{8 n + 1 \cdot 4} a^{7 n - 12}$

Multiply $4$ by $1$ .

$34 a^{8 n + 4} a^{7 n - 12}$

Multiply $a^{8 n + 4}$ by $a^{7 n - 12}$ by adding the exponents.

Move $a^{7 n - 12}$ .

$34 (a^{7 n - 12} a^{8 n + 4})$

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

$34 a^{7 n - 12 + 8 n + 4}$

Add $7 n$ and $8 n$ .

$34 a^{15 n - 12 + 4}$

Add $- 12$ and $4$ .

$34 a^{15 n - 8}$