Exponent of Zero and Negative Exponents - Purdue University

[Pages:5]16-week Lesson 2 (8-week Lesson 1)

Exponent of Zero and Negative Exponents

When an exponent is a positive integer, such as 1, 2, 3, 4, ... , exponential

notation represents the product of repeated factors (the base times itself

some number of times) o 2 = the exponent of 2 indicates there are 2 factors of o 5 = the exponent of 5 indicates there are 5 factors of o = ... the exponent of indicates there are factors of

What about when an exponent is not a positive integer? In this section we'll look at exponents of zero and exponents that are negative integers.

One way to approach exponents of zero is to think about a term divided by

itself;

for

instance,

2 2

=

1

because

anything

over

itself

is

one.

However,

what

happens

if

we

simplified

2 2

using

the

Quotient

Rule

that

was

discussed earlier?

2 2

=

2-2

=

0

This

shows

that

2 2

=

0,

and

since

we

already

know

that

2 2

=

1,

that

means 0 must equal 1. This leads us to the Zero-Exponent Rule.

Zero-Exponent Rule:

- any base taken to the power of zero is 1

o

the

exception

to

this

rule

is

a

base

of

zero,

because

using

the

2 2

example, you cannot have a denominator of zero

- this is true for a factor like 0 = 1, as well as a product like

(576)0 = 1 or a quotient like (-4392)0 = 1

1

16-week Lesson 2 (8-week Lesson 1)

o 0 =

Exponent of Zero and Negative Exponents

()0 =

o -40 =

(-4)0 =

The final topic in this lesson is negative exponents. Our goal when

working with negative exponents is to make them positive, since we have

already covered exponent rules with positive integers. One way to

understand how to change a negative exponent to a positive exponent is to

think about canceling common factors within a fraction. For instance,

2 3

=

,

and

since

this

fraction

has

common

factors

in

the

numerator

and

denominator, we can simply cancel two factors of from both to get 1.

However, what happens if we simplify

2 3

using

the

Quotient

Rule?

2 3

=

2-3

=

-1

So

what

we

see

is

that

2 3

simplifies

to

both

-1

and

1;

and

since

-1

and

1

are

both

equal

to

23,

they

are

also

equal

to

each

other.

So this shows that

to change the sign of an exponent, we can simply take the reciprocal of the

factor that has a negative exponent.

2

16-week Lesson 2 (8-week Lesson 1)

Exponent of Zero and Negative Exponents

Negative Exponent Rule:

- to change the sign of an exponent, take the reciprocal of the

expression or factor with the negative exponent

o

-2

=

11 2 (-3)2

-5

=

1 5

=

5

notice we do not take the reciprocal of the exponent, but

rather the factor that contains a negative exponent

- remember that when an exponent is a positive integer, exponential

notation represents the product of repeated factors (something times

itself times some number of times)

- the sign of the base does NOT change

o

(-2)-4

=

1 (-3)2

-2-4 =

=

1 (-3)(-3)

- again, this is true for a factor or a product/quotient

o Product to a Power (-24)-3 =

Quotient to a Power (3-332)-4 =

3

16-week Lesson 2 (8-week Lesson 1)

Exponent of Zero and Negative Exponents

Example 1: Simplify each expression COMPLETELY. Do NOT leave

negative exponents in your answers.

a. -82(33)-4

b.

-8

2

1 (33)4

b. (-8)2(3-3)-4

(-8)2()2(3)-4(-3)-4

-82 8112

64

2

1 34

12

-

642 1 12 1 81 1

-8 8110

c.

(12 4-3)5

(-

5 7

0

-)

d.

-82

8112

-82 8112

-82 8112

-82 8112

e.

-3-1(6-4)2

(40-3 1)-3

(1

2

5

-43)

f.

-1

1 3

(64)2

(414

-3

)

(234)5

-

1 3

36 8

(44)3

1

15 3220

-

1 3

36 8

6412 1

15 3220

-

366427 33228

-

d. (2-47)3 (-2-65)2

(247)3 (2-516)2

8

12

21

4

1 1012

81221 41012

f. (-27-4)3 (-256)-2

4

16-week Lesson 2 (8-week Lesson 1)

g.

(1

3

-5 2 )-1

(923)-2

h.

i. -32 + 70 - 2-1

Exponent of Zero and Negative Exponents

h.

(12-30)-1 -2

(12)-1(-3)-1(0)-1 -2

23(1) -2

232

j.

-0

+

4-2

+

(16)-1

7

-1

0

+

1 42

+

(

7

1

)

16

-1

1

+

1 16

+

7 16

-1 + 1 + 7

16 16

- 16 + 1 + 7

16 16 16

-

Answers to Examples:

1a.

-8 8110

;

1b.

6414 ; 1c.

81

20 3215

;

1d.

229 ; 1e.

-2427 28

;

1f.

-2 2411

;

1g.

278

;

1h.

222 ; 1i.

-

17 2

;

1j.

-1

2

5

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download