This document discusses multiplying monomials and powers. It defines a monomial as a number, variable, or the product of a number and variable with no division. To multiply monomials with the same base, add the exponents. When multiplying terms with the same variable base but different coefficients, multiply the coefficients and add the exponents. Powers can be multiplied by multiplying the exponents, and powers can be raised to another power by multiplying those exponents.

1. 8-1 Multiplying Monomials (sounds like some sort of disease, doesn’t it???)

2. 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!)

3. Determine if the following are monomials: -3x 2 y 11 3m + 4n xyz 4h / 3j

4. The parts of a monomial coefficient 3m² exponent base

5. Product of POWERS To multiply two powers that have the same base, ADD the exponents: m ² • m³ = m 5 To multiply two monomials that have the same base, with coefficients: multiply BIG, add LITTLE: ( 5x )( 2x ² ) = 10x ³ *Don’t forget that variables without an exponent are understood to have a power of 1!!

6. Let’s try something harder! How about this one?

7. 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: x 5 y 6 SQUASH THEM TOGETHER, AND YOUR ANSWER IS -6 x 5 y 6

8. Power of a Power To raise a power to a power, you MULTIPLY the exponents: (x ³)² = x 6 If there is a constant involved, don’t forget to raise it to the power as well! (2m²) 4 = 16m 8

9. Power of a Product: Raise each factor to that same power (2x 3 y 4 ) 5 = 32x 15 y 20 (now that’s POWERFUL!)

10. Putting it all together: Simplify the following, using the rules we have just covered: 2 x 5 y 4 (2 x 3 y 6 ) 5 (4 x 2 y ) (2 xy 2 z 3 ) 3

11. Applications GEOMETRY: Express the area of this circle as a monomial. Area = π r 2 (Formula for the area of a circle)

12. More applications Find the volume of the rectangular solid: Volume of a rectangular solid: l •w•h