Algebra 1Ch 8.2 – Zero & Negative Exponents

Objective 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… In this lesson we will extend and use what

we learned to include zero exponents and negative exponents…

Let’s look at the rules…

Zero Exponents RULE: a nonzero number raised to the

zero power is equal to 1

Example:

a0 = 1 when, a ≠ 0

Reciprocals 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:

2

6 2

6

Original ReciprocalExample: Product

● = 1

Negative Exponents Rule: a-n is the reciprocal of an

when, a ≠ 0

Example:

a-n =1

an

Examples Powers with negative & zero exponents

2 2 1

22

1

4

(-2)0 = 1

5 x 1

5x

1

3

1FHGIKJ

3

0 3 1

03

a

b

c

d

e

Undefined – zero has no reciprocal!

Simplifying Expressions Ok…now that you know the rules…let’s

look at simplifying some expressions… Before we do that… be forewarned… you

need to know how to work with fractions here!

Reminder - when multiplying fractions you multiply the numerators and you multiply the denominators

Example #1 Rewrite with positive exponents: 5(2-x)

51

2 x

When analyzing this expression I see that it has a negative exponent.

Don’t forget that a whole number written as a fraction is the number over 1

I will need to write the reciprocal of 2-x before I multiply by 5.

Solution:5(2-x)

5

2 x

Example #2 Rewrite with positive exponents

21 1

2 3x y

2x-2y-3

When analyzing this expression I see that it has negative exponents.

I will need to write them as reciprocals before I multiply

Solution:

2x-2y-3 22 3x y

Evaluating Expressions Ok…now that you know how to simplify an

expression…Let’s look at evaluating expressions…

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…

Again…the key is to analyze the expression first…

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)

1

33

22

3

3

2

2 9

9

Evaluate the expression 3-2 ● 32

Solution #1:

3-2 ● 32 = 3-2 + 2 = 30 = 1

3-2 ● 32

Solution #2:

1

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…

Again, you will use what you learned about zero and negative exponents and the multiplication properties of exponents…

The key is to analyze the expression first…

Example #5

1

5

12 2a

Rewrite with positive exponents (5a)-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:

(5a)-2 = 5-2 ● a-2 1

25 2a

Example #6

Rewrite with positive exponents

13d n

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 -3n

Therefore, using the definition of a negative exponent I can rewrite the expression as:

(d-3n)-1

Example #6 (Continued)

(d-3n)-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-3n)-1 = d(-3n)●(-1) = d3n

Comments On the next couple of slides are some practice

problems…The answers are on the last slide…

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…

If you cannot find the error bring your work to me and I will help…

Your Turn

1.

2.

3.

4.

5.

4-2

1

5

1FHGIKJ

4(4-2)

2-3 ● 22

(-3-2)-9

Evaluate the exponential expression. Write fractions in simplest form

Your Turn

1

4 10 14x y

6.

7.

8.

9.

10.

Rewrite the expression with positive exponents

x-5

8x-2y-6

(-10a)0

FHG

IKJ

4

2

2

1

1x

x