Simplify -3mn(m^2n^3+2mn)

$- 3 m n (m^{2} n^{3} + 2 m n)$

Step 1

Apply the distributive property.

$- 3 m n (m^{2} n^{3}) - 3 m n (2 m n)$

Step 2

Multiply $m$ by $m^{2}$ by adding the exponents.

Move $m^{2}$ .

$- 3 (m^{2} m) n \cdot n^{3} - 3 m n (2 m n)$

Multiply $m^{2}$ by $m$ .

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

$- 3 (m^{2} m^{1}) n \cdot n^{3} - 3 m n (2 m n)$

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

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

Add $2$ and $1$ .

$- 3 m^{3} n \cdot n^{3} - 3 m n (2 m n)$

Step 3

Multiply $m$ by $m$ by adding the exponents.

Move $m$ .

$- 3 m^{3} n \cdot n^{3} - 3 (m \cdot m) n (2 n)$

Multiply $m$ by $m$ .

$- 3 m^{3} n \cdot n^{3} - 3 m^{2} n (2 n)$

Step 4

Simplify each term.

Multiply $n$ by $n^{3}$ by adding the exponents.

Move $n^{3}$ .

$- 3 m^{3} (n^{3} n) - 3 m^{2} n (2 n)$

Multiply $n^{3}$ by $n$ .

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

$- 3 m^{3} (n^{3} n^{1}) - 3 m^{2} n (2 n)$

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

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

Add $3$ and $1$ .

$- 3 m^{3} n^{4} - 3 m^{2} n (2 n)$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$- 3 m^{3} n^{4} - 3 m^{2} (n \cdot n) \cdot 2$

Multiply $n$ by $n$ .

$- 3 m^{3} n^{4} - 3 m^{2} n^{2} \cdot 2$

Multiply $2$ by $- 3$ .

$- 3 m^{3} n^{4} - 6 m^{2} n^{2}$