8 2 notes zero and negative exponents -...

• I can evaluate zero and negative exponents.• I can graph and solve problems with exponential functions.

Algebra 8.2 Notes Zero and Negative Exponents(pp 456-458) of 9

456 Chapter 8 Exponents and Exponential Functions

Zero and Negative Exponents

USING ZERO AND NEGATIVE EXPONENTS

In this lesson you will study how to use the multiplication properties ofexponents when working with negative exponents.

Powers with Zero and Negative Exponents

a. 2º2 = = !14! 2º2 is the reciprocal of 22.

b. (º2)0 = 1 a0 is 1.

c. 5ºx = !51x! 5ºx is the reciprocal of 5x.

d. !!13!"º1

= 3 The reciprocal of }13} is 3.

e. 0º3 = (Undefined) Zero has no reciprocal.1!03

1!22

E X A M P L E 1

GOAL 1

Evaluate powersthat have zero and negativeexponents.

Graph exponentialfunctions.

! To solve real-life problemssuch as predicting a player’saverage score per game inExample 6.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

8.2

RE

AL LIFE

RE

AL LIFE

Let a be a nonzero number and let n be a positive integer.

• A nonzero number to the zero power is 1: a0 = 1, a ! 0.

• a–n is the reciprocal of an: aºn = !a1n!, a ! 0.

DEFINITION OF ZERO AND NEGATIVE EXPONENTS

Investigating Zero and Negative ExponentsCopy the table and discuss any patterns you see. Use the patterns tocomplete the table. Write non-integers as fractions in simplest form.

What appears to be the value of a0 for any number a?

How can you evaluate an expression of the form aºn?3

2

1

DevelopingConcepts

ACTIVITY

Exponent, n 3 2 1 0 º1 º2 º3

Power, 2n 8 4 2 ? ? ? ?

Power, 3n 27 9 3 ? ? ? ?

Power, 4n 64 16 4 ? ? ? ?

HOMEWORK HELPVisit our Web site

www.mcdougallittell.comfor extra examples.

INTE

RNET

STUDENT HELP


GRAPHING EXPONENTIAL FUNCTIONS

So far we have used expressions of the form bn where b ! 0 and n is an integer.To model some situations, we need an of the form y = a • bx where x is a real number, b > 0, and b ! 1. To graph the exponentialfunction y = a • bx, make a table using integer values for x and plot thecorresponding points. To complete the graph for all real values of x, connect thepoints with a smooth curve.

Graphing an Exponential Function

To sketch the graph of y = 2x, make a table that includes negative x-values.

Draw a coordinate plane and plot the sixpoints given by the table. Then draw asmooth curve through the points.

Notice that the graph has a y-intercept of 1, and that it gets closer to the negativeside of the x-axis as the x-values getsmaller.

. . . . . . . . . .

In real-life applications of y = abx, x is often the time period.

Evaluating an Exponential Function

A professional basketball player’s first year in the NBA was 1998. Suppose itwere estimated that from 1998 to 2010 his average points per game could bemodeled by P = 18.5(1.038)t, where t = 0 represents the year 2000.

a. Estimate the player’s average points per game in 1998.

b. Estimate his average points per game in the year 2000.

SOLUTION

a. P = 18.5 • 1.038º2 Substitute º2 for t.

" 17.17 Use a calculator.

! In the year 1998 his average points per game was about 17.2 points.

b. P = 18.5 • 1.0380 Substitute 0 for t.

= 18.5 a0 is 1.


E X A M P L E 6

E X A M P L E 5

exponential function

GOAL 2

!1!3

7

5

y

x31

3

(3, 8)

(2, 4)

y ! 2x

(1, 2)(0, 1)

!!2, "14

!!1, "12

RE

AL LIFE

RE

AL LIFE

Basketball

x º2 –1 0 1 2 3

2x 2º2 = "14" 2º1 = "

12" 20 = 1 2 4 8








. . . . . . . . . .






SOLUTION





= 18.5 a0 is 1.


E X A M P L E 6

E X A M P L E 5


GOAL 2

!1!3

7

5

y

x31

3

(3, 8)

(2, 4)

y ! 2x

(1, 2)(0, 1)

!!2, "14

!!1, "12

RE

AL LIFE

RE

AL LIFE

Basketball

x º2 –1 0 1 2 3

2x 2º2 = "14" 2º1 = "

12" 20 = 1 2 4 8

Evaluate the following:


1. 2.

3. 4.

5.



Zero and Negative Exponents

USING ZERO AND NEGATIVE EXPONENTS

In this lesson you will study how to use the multiplication properties ofexponents when working with negative exponents.

Powers with Zero and Negative Exponents

a. 2º2 = = !14! 2º2 is the reciprocal of 22.

b. (º2)0 = 1 a0 is 1.

c. 5ºx = !51x! 5ºx is the reciprocal of 5x.

d. !!13!"º1

= 3 The reciprocal of }13} is 3.

e. 0º3 = (Undefined) Zero has no reciprocal.1!03

1!22

E X A M P L E 1

GOAL 1

Evaluate powersthat have zero and negativeexponents.

Graph exponentialfunctions.

! To solve real-life problemssuch as predicting a player’saverage score per game inExample 6.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

8.2

RE

AL LIFE

RE

AL LIFE

Let a be a nonzero number and let n be a positive integer.

• A nonzero number to the zero power is 1: a0 = 1, a ! 0.

• a–n is the reciprocal of an: aºn = !a1n!, a ! 0.

DEFINITION OF ZERO AND NEGATIVE EXPONENTS

Investigating Zero and Negative ExponentsCopy the table and discuss any patterns you see. Use the patterns tocomplete the table. Write non-integers as fractions in simplest form.

What appears to be the value of a0 for any number a?

How can you evaluate an expression of the form aºn?3

2

1

DevelopingConcepts

ACTIVITY

Exponent, n 3 2 1 0 º1 º2 º3

Power, 2n 8 4 2 ? ? ? ?

Power, 3n 27 9 3 ? ? ? ?

Power, 4n 64 16 4 ? ? ? ?

HOMEWORK HELPVisit our Web site

www.mcdougallittell.comfor extra examples.

INTE

RNET

STUDENT HELP

8.2 Zero and Negative Exponents 457

Simplifying Exponential Expressions

Rewrite with positive exponents.a. 5(2ºx) b. 2xº2yº3

SOLUTION

a. 5(2ºx) = 5!!21x!" = !

25x! b. 2xº2yº3 = 2 • • =

Evaluating Exponential Expressions

Evaluate the expression.

a. 3º2 • 32 b. (2º3)º2 c. 3º4

SOLUTION

a. 3º2 • 32 = 3º2 + 2 Use product of powers property.

= 30 Add exponents.

= 1 a0 is 1.

b. (2º3)º2 = 2º3 • (º2) Use power of a power property.

= 26 Multiply exponents.

= 64 Evaluate.

c. You might want to evaluate 3º4 with a calculator.

KEYSTROKES DISPLAY

3 4


Rewrite with positive exponents.

a. (5a)º2 b. !dº

13n!

SOLUTION

a. (5a)º2 = 5º2 • aº2 Use power of a product property.

= !512! • !

a12! Write reciprocals of 52 and a2.

= !25

1a2! Multiply fractions.

b. = (dº3n)º1Use definition of negative exponents.

= d(º3n) • (º1) Use power of a power property.

= d3n Multiply exponents.

1!dº3n

E X A M P L E 4

E X A M P L E 3

2!x2y3

1!y3

1!x2

E X A M P L E 2

.01234568

STUDENT HELP

Study Tip Informally, you can thinkof rewriting expressionswith positive exponentsas “moving factors” fromthe denominator to thenumerator or vice versa.

!3x

1º4! = !

x3

4!

STUDENT HELP

KEYSTROKE HELP

Use or to input the

exponent º4.




SOLUTION

a. 5(2ºx) = 5!!21x!" = !

25x! b. 2xº2yº3 = 2 • • =



a. 3º2 • 32 b. (2º3)º2 c. 3º4

SOLUTION


= 30 Add exponents.

= 1 a0 is 1.



= 64 Evaluate.


KEYSTROKES DISPLAY

3 4



a. (5a)º2 b. !dº

13n!

SOLUTION


= !512! • !


= !25





1!dº3n

E X A M P L E 4

E X A M P L E 3

2!x2y3

1!y3

1!x2

E X A M P L E 2

.01234568

STUDENT HELP


!3x

1º4! = !

x3

4!

STUDENT HELP

KEYSTROKE HELP

Use or to input the

exponent º4.

6. 7.

8. 9.





SOLUTION

a. 5(2ºx) = 5!!21x!" = !

25x! b. 2xº2yº3 = 2 • • =



a. 3º2 • 32 b. (2º3)º2 c. 3º4

SOLUTION


= 30 Add exponents.

= 1 a0 is 1.



= 64 Evaluate.


KEYSTROKES DISPLAY

3 4



a. (5a)º2 b. !dº

13n!

SOLUTION


= !512! • !


= !25





1!dº3n

E X A M P L E 4

E X A M P L E 3

2!x2y3

1!y3

1!x2

E X A M P L E 2

.01234568

STUDENT HELP


!3x

1º4! = !

x3

4!

STUDENT HELP

KEYSTROKE HELP

Use or to input the

exponent º4.




SOLUTION

a. 5(2ºx) = 5!!21x!" = !

25x! b. 2xº2yº3 = 2 • • =



a. 3º2 • 32 b. (2º3)º2 c. 3º4

SOLUTION


= 30 Add exponents.

= 1 a0 is 1.



= 64 Evaluate.


KEYSTROKES DISPLAY

3 4



a. (5a)º2 b. !dº

13n!

SOLUTION


= !512! • !


= !25





1!dº3n

E X A M P L E 4

E X A M P L E 3

2!x2y3

1!y3

1!x2

E X A M P L E 2

.01234568

STUDENT HELP


!3x

1º4! = !

x3

4!

STUDENT HELP

KEYSTROKE HELP

Use or to input the

exponent º4.

Evaluate the expression.10. 11.

12.





SOLUTION

a. 5(2ºx) = 5!!21x!" = !

25x! b. 2xº2yº3 = 2 • • =



a. 3º2 • 32 b. (2º3)º2 c. 3º4

SOLUTION


= 30 Add exponents.

= 1 a0 is 1.



= 64 Evaluate.


KEYSTROKES DISPLAY

3 4



a. (5a)º2 b. !dº

13n!

SOLUTION


= !512! • !


= !25





1!dº3n

E X A M P L E 4

E X A M P L E 3

2!x2y3

1!y3

1!x2

E X A M P L E 2

.01234568

STUDENT HELP


!3x

1º4! = !

x3

4!

STUDENT HELP

KEYSTROKE HELP

Use or to input the

exponent º4.

Rewrite with positive Exponents. Assume n is positive.13. 14.

15. 16.









. . . . . . . . . .






SOLUTION





= 18.5 a0 is 1.


E X A M P L E 6

E X A M P L E 5


GOAL 2

!1!3

7

5

y

x31

3

(3, 8)

(2, 4)

y ! 2x

(1, 2)(0, 1)

!!2, "14

!!1, "12

RE

AL LIFE

RE

AL LIFE

Basketball

x º2 –1 0 1 2 3

2x 2º2 = "14" 2º1 = "

12" 20 = 1 2 4 8

17. Example: Sketch the graph of .

Sketch the graph of y = 12

⎛⎝⎜

⎞⎠⎟x









. . . . . . . . . .






SOLUTION





= 18.5 a0 is 1.


E X A M P L E 6

E X A M P L E 5


GOAL 2

!1!3

7

5

y

x31

3

(3, 8)

(2, 4)

y ! 2x

(1, 2)(0, 1)

!!2, "14

!!1, "12

RE

AL LIFE

RE

AL LIFE

Basketball

x º2 –1 0 1 2 3

2x 2º2 = "14" 2º1 = "

12" 20 = 1 2 4 8

Census figures showed that a town’s population can be modeled by where t = 0 represents the year 2010.

18. Estimate the population in the year 1999.

19. Estimate the population in the year 2010.

A middle range sales price for a house in the U.S. can be modeled by where t represents the year 1997 and P is the price in thousands of dollars.

20. Estimate the middle home sales price in the year 1991.

21. Estimate the price in the year 1997.

22. The function is a(n) _?__ function.

23. Describe the error.

24. Does the graph appear to be the graph of the exponential function ?


25. ____ True or False: If a is positive, then is positive. Explain your reasoning.


8 2 notes zero and negative exponents -...

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