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
base
exponent
Product of POWERS
To multiply two monomials that have the same base, with coefficients: multiply BIG, add LITTLE: To multiply two powers that have the same base, ADD the exponents:
( 5x )( 2x ² ) = 10x³
*Don’t forget that variables without an exponent are understood to have a power of 1!!
m² • m³ = m5
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 = - 6
THEN, 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
Applications
GEOMETRY: Express the area of this circle as a monomial.
Area = πr 2 (Formula for the area of a circle)
More applications
Find the volume of the rectangular solid:
Volume of a rectangular solid: l•w•h