Simplify 5(x^(2n-1))*(2x^(3n+1))^2

$5 (x^{2 n - 1}) \cdot {(2 x^{3 n + 1})}^{2}$

Step 1

Apply the product rule to $2 x^{3 n + 1}$ .

$5 x^{2 n - 1} \cdot (2^{2} {(x^{3 n + 1})}^{2})$

Step 2

Rewrite using the commutative property of multiplication.

$5 \cdot 2^{2} x^{2 n - 1} {(x^{3 n + 1})}^{2}$

Step 3

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

$5 \cdot 4 x^{2 n - 1} {(x^{3 n + 1})}^{2}$

Step 4

Multiply $5$ by $4$ .

$20 x^{2 n - 1} {(x^{3 n + 1})}^{2}$

Step 5

Multiply the exponents in ${(x^{3 n + 1})}^{2}$ .

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

$20 x^{2 n - 1} x^{(3 n + 1) \cdot 2}$

Apply the distributive property.

$20 x^{2 n - 1} x^{3 n \cdot 2 + 1 \cdot 2}$

Multiply $2$ by $3$ .

$20 x^{2 n - 1} x^{6 n + 1 \cdot 2}$

Multiply $2$ by $1$ .

$20 x^{2 n - 1} x^{6 n + 2}$

Step 6

Multiply $x^{2 n - 1}$ by $x^{6 n + 2}$ by adding the exponents.

Move $x^{6 n + 2}$ .

$20 (x^{6 n + 2} x^{2 n - 1})$

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

$20 x^{6 n + 2 + 2 n - 1}$

Add $6 n$ and $2 n$ .

$20 x^{8 n + 2 - 1}$

Subtract $1$ from $2$ .

$20 x^{8 n + 1}$