8-1 Multiplying Monomials

(sounds like some sort of disease, doesn’t it???)

What is a MONOMIAL?

A monomial can be defined as:

a number (by itself, known as a constant) a variable, or the product of a number and a variable

(any expression involving the DIVISION of variables is NOT a monomial!)

Determine if the following are monomials:

-3x 2y

11

3m + 4n

xyz

4h / 3j

The parts of a monomial

coefficient

3m² exponent

base

Product of POWERS

To multiply two powers that have the same base, ADD the exponents:

m² • m³ = m5

To multiply two monomials that have the same base, with coefficients: multiply BIG, add LITTLE:

( 5x )( 2x ² ) = 10x³

*Don’t forget that variables without an exponent are understood to have a power of 1!!

Let’s try something harder!

How about this one?

DON’T PANIC!!

JUST FOLLOW THE RULES AND GO ONE STEP AT A TIME!

FIRST, MULTIPLY ALL OF THE COEFFICIENTS TOGETHER:

- 5 · 3 · 2/5 = - 6THEN, ADD UP THE EXPONENTS ON THE VARIABLES:

THERE ARE X’S AND Y’S TO COUNT UP: HOW MANY X’S ARE THERE? HOW MANY Y’S ARE THERE?

YOU SHOULD GET: x5y6

SQUASH THEM TOGETHER, AND YOUR ANSWER IS -6 x5y6

Power of a Power

To raise a power to a power,

you MULTIPLY the exponents:

(x³)² = x6

If there is a constant involved,

don’t forget to raise it to the power as well!

(2m²)4 = 16m8

Power of a Product:

Raise each factor to that same power

(2x3y4)5 = 32x15y20

(now that’s POWERFUL!)

Putting it all together:Simplify the following,

using the rules we have just covered:

2x5y4(2x3y6)5

(4x2y) (2xy2z3)3

ApplicationsGEOMETRY: Express the area

of this circle as a monomial.

Area = πArea = πr r 22 (Formula for the area of a circle) (Formula for the area of a circle)

More applicationsFind the volume of the rectangular solid:

Volume of a rectangular solid: l•w•h