Properties of Exponents II Product of Monomials Quotient of Monomials

Review: Exponential Notation Exponential notation serves as a shorthand to keep us from having to write out a variable or number over and over and over and…. The exponent of the base tells us how many times that base is multiplied by itself.

Review: Negative Exponents Negative exponents should never be a part of your answer. Negative exponents move the base to other part of the fraction. Remember to simplify that base as much as possible before you move it. When you move the base, the exponent becomes positive.

Monomials The prefix “mono” means “one.” In this case, we are saying “one term” Monomials are a product of numbers and variables.

Monomials Determine if the following are monomials. 5x 3 YES 8x 2 y 3 z 4 YES 5a 3 -2b 2 +3c NO (This is a TRInomial  3 terms)

Products of Monomials We will look at this through an example. Ex. a 2  a 3  b 3  b 2  a 4 Remember that exponential notation tells us how many times the base is multiplied by itself. That means this expression can be written as: (a  a)  (a  a  a)  (b  b  b)  (b  b)  (a  a  a  a)

Products of Monomials Looking at that last expression, we can bring back exponential notation and get: a 9 b 5 We can express this as: a n  a m =a n+m

Products of Monomials When trying to simplify an expression, it may help to consider how many of a particular base are represented. Let’s look at a couple of examples:

Products of Monomials x 3  x 6  y 2  x  y 10 I am thinking to myself, there are 3 x’s in the first factor, 6 x’s in the second, and 1 x in the fourth. All total, there are 10 x’s. I also say to myself that there are 2 y’s in the second factor and 10 y’s in the fifth. All total, there are 12 of them. That means my final answer is x 10 y 12.

Products of Monomials Make no mistake, we are still multiplying. We just need to consider a different task when we are carrying out this multiplication. The point is, when you are multiplying monomials, you add the exponents.

Products of Monomials Refer to your notes for examples done in class.

Quotient of Monomials The important point to keep in mind here, we are simplifying fractions. Simplify the numerical part of the fractions first.

Quotient of Monomials Let’s look at what is actually happening. To do this, we again use exponential notation. See your notes for the example in class.

Quotient of Monomials The net result of looking at exponential notation is when your simplifying a fraction: You subtract the exponents and keep the base wherever the exponent was higher.

Quotient of Monomials Refer to your notes for examples from class.