Algebra 1 Ch 8 2 Zero Negative Exponents

Algebra 1 Ch 8. 2 – Zero & Negative Exponents

Objective n Students will evaluate powers that have zero and negative exponents

Before we begin… In the last lesson we looked at the multiplication properties of exponents… n In this lesson we will extend and use what we learned to include zero exponents and negative exponents… n Let’s look at the rules… n

Zero Exponents n RULE: a nonzero number raised to the zero power is equal to 1 Example: a 0 = 1 when, a ≠ 0

Reciprocals n n n When working with negative exponents you need to know what a reciprocal is… We already covered this earlier in the course so as a quick review… A reciprocal is a fraction that is inverted and the product is 1. It looks like this: Example: Original 2 6 Reciprocal ● 6 2 Product = 1

Negative Exponents n Rule: a-n is the reciprocal of an Example: a-n = 1 an when, a ≠ 0

Examples n Powers with negative & zero exponents d a b c (-2)0 = 1 e Undefined – zero has no reciprocal!

Simplifying Expressions Ok…now that you know the rules…let’s look at simplifying some expressions… n Before we do that… be forewarned… you need to know how to work with fractions here! n Reminder - when multiplying fractions you multiply the numerators and you multiply the denominators n

Example #1 n Rewrite with positive exponents: 5(2 -x) When analyzing this expression I see that it has a negative exponent. I will need to write the reciprocal of 2 -x before I multiply by 5. Don’t forget that a whole number written as a fraction is the number over 1 Solution: 5(2 -x)

Example #2 n Rewrite with positive exponents 2 x-2 y-3 When analyzing this expression I see that it has negative exponents. I will need to write them as reciprocals before I multiply Solution: 2 x-2 y-3

Evaluating Expressions Ok…now that you know how to simplify an expression…Let’s look at evaluating expressions… n You will use what you learned in this lesson about zero and negative exponents and combine that with what you learned about the multiplication properties of exponents… n Again…the key is to analyze the expression first… n

Example #3 Evaluate the expression 3 -2 ● 32 When analyzing this expression I see that I have a negative exponent. But I also see that I multiplying 2 powers with the same base… I have to make a decision here…either I work with the negative exponent first or I work with the product of powers property…either way I will get the same answer… If I work with the negative exponents first…. it will take me more steps to get to the answer…so I choose to work with the product of powers property, which states when multiplying powers if the base is the same add the exponents…(We will look at both solutions)

Example #3 (Continued) Evaluate the expression 3 -2 ● 32 Solution #1: 3 -2 ● 32 = 3 -2 + 2 = 30 = 1 Solution #2: 3 -2 ● 32

Example #4 Evaluate the expression (2 -3)-2 When analyzing this expression I see that I have 2 negative exponents. I also see that I can use the Power of a Power Property, which states, to find the power of a power, multiply the exponents. Solution: (2 -3)-2 = 2 -3●(-2) = 26 = 64

Simplifying Exponential Expressions In this section we will simplify exponential expressions, that is…we will write the expressions with positive exponents… n Again, you will use what you learned about zero and negative exponents and the multiplication properties of exponents… n The key is to analyze the expression first… n

Example #5 Rewrite with positive exponents (5 a)-2 When analyzing this expression I see that I can use the Power of a Product Property, which states to find the power of a product, find the power of each factor and multiply Solution: (5 a)-2 = 5 -2 ● a-2

Example #6 Rewrite with positive exponents This example is a little harder and requires some higher order thinking skills…. First, I need to recognize that this expression is the reciprocal of some other expression…how I recognize that is I see that it is 1 over the expression d -3 n Therefore, using the definition of a negative exponent I can rewrite the expression as: (d-3 n)-1

Example #6 (Continued) (d-3 n)-1 Now that the expression is in a format that is not fraction form…I see that I can use the Power of a Power Property, which states to find the power of a power, multiply the exponents Solution: (d-3 n)-1 = d(-3 n)●(-1) = d 3 n

Comments n On the next couple of slides are some practice problems…The answers are on the last slide… n Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… n If you cannot find the error bring your work to me and I will help…