# How do you find the product 4km^2(8km^2+2k^2m+5k)?

May 29, 2018

4km^2(8km^2 + 2k^2m + 5k)=color(blue)(32k^2m^4 + 8k^3m^3 + 20k^2m^2

#### Explanation:

Simplify:

$4 k {m}^{2} \left(8 k {m}^{2} + 2 {k}^{2} m + 5 k\right)$

Distribute $4 k {m}^{2}$ by multiplying by each of the terms in parentheses.

$\left(4 k {m}^{2} \cdot 8 k {m}^{2}\right) + \left(4 k {m}^{2} \cdot 2 {k}^{2} m\right) + \left(4 k {m}^{2} \cdot 5 k\right)$

Recombine the constants and variables.

$\left(4 \cdot 8 \cdot k \cdot k \cdot {m}^{2} \cdot {m}^{2}\right) + \left(4 \cdot 2 \cdot k \cdot {k}^{2} \cdot {m}^{2} \cdot m\right) + \left(4 \cdot 5 \cdot k \cdot k \cdot {m}^{2}\right)$

Multiply the constants.

$\left(32 \cdot k \cdot k \cdot {m}^{2} \cdot {m}^{2}\right) + \left(8 \cdot k \cdot {k}^{2} \cdot {m}^{2} \cdot m\right) + \left(20 \cdot k \cdot k \cdot {m}^{2}\right)$

Apply product rule: ${a}^{m} {a}^{n} = {a}^{m + n}$

Recall that no exponent is understood to be $1$.

$\left(32 \cdot {k}^{1 + 1} {m}^{2 + 2}\right) + \left(8 \cdot {k}^{1 + 2} \cdot {m}^{2 + 1}\right) + \left(20 \cdot {k}^{1 + 1} \cdot {m}^{2}\right)$

Simplify.

$32 {k}^{2} {m}^{4} + 8 {k}^{3} {m}^{3} + 20 {k}^{2} {m}^{2}$