# How do you multiply monomials by monomials?

Mar 27, 2018

$\implies {a}_{1} {x}^{{p}_{1}} \cdot {a}_{2} {x}^{{p}_{2}} = {a}_{1} {a}_{2} {x}^{{p}_{1} + {p}_{2}}$

#### Explanation:

A monomial is of the form:

$\implies a {x}^{p}$

where $a$ is a constant coefficient and $p$ is a constant power.

In the case of multiplying two monomials together:

$\implies A {x}^{P} \equiv {a}_{1} {x}^{{p}_{1}} \cdot {a}_{2} {x}^{{p}_{2}}$

The coefficients will multiply, so:

$\implies A = {a}_{1} \cdot {a}_{2}$

The powers will sum, so:

$\implies P = {p}_{1} + {p}_{2}$

Hence:

$\implies A {x}^{P} \equiv {a}_{1} {x}^{{p}_{1}} \cdot {a}_{2} {x}^{{p}_{2}} = {a}_{1} {a}_{2} {x}^{{p}_{1} + {p}_{2}}$

For example:
$\implies 3 {x}^{2} \cdot 2 x$

$\implies \left(3 \cdot 2\right) {x}^{2 + 1}$

$\implies 6 {x}^{3}$

Mar 27, 2018

Multiply all the numbers and variables together (use the Product of Powers Rule for exponents) and simplify.

#### Explanation:

Here's an example:
$2 {x}^{2} {y}^{4} z \cdot 4 {a}^{3} {x}^{3} {z}^{3}$
We see that we have two numbers, two $x$'s, one $a$, one $y$, and two $z$'s. We can use the Product of Powers Rule to simply add the exponents for the $x$'s and $z$'s. $2 \cdot 4 = 8 , {x}^{2} \cdot {x}^{3} = {x}^{5} , \mathmr{and} z \cdot {z}^{3} = {z}^{4}$. So $2 {x}^{2} {y}^{4} z \cdot 4 {a}^{3} {x}^{3} {z}^{3} = 8 {a}^{3} {x}^{5} {y}^{4} {z}^{4}$.