6.1 using integers as exponents bobsmathclass.com copyright © 2010 all rights reserved. 1 if m and...

1

6.1 Using Integers as Exponents

BobsMathClass.Com Copyright © 2010 All Rights Reserved.

5 7 121. 15x 3x 45x

75 352. x x

48 3 32 123. x y x y

If m and n are positive integers and a and b are real numbers (b0 when it is a denominator), then:

Let’s review some properties of exponents where the exponents were always positive integers.

n m n m1. b b b

mn nm2. b b

n n n3. ab a b

Examples:Properties:

nn m

mb

5. i) b if n mb

n

m m nb 1

ii) if n<mb b

n n

na a

4. b b

3 3

3x x

4. y y

In this example, the expression cannot be simplified because the bases are different.

Next Slide

83

5x

5. xx

5

8 3x 1

x x

If the bases are the same, subtract the smaller exponent from the larger exponent. Keep the variable where the exponent is larger.

2



We will now introduce how to work with negative exponents. Before we give a definition, let’s experiment with our calculators. 3Evaluate: 2

Directions:

Type 2 then the exponent button. The exponent button will either be xy or .Then type in –3. Use the (+-) button to make the 3 negative. You should see 0.125 on the calculator.

0.12518

Notice the result of a positive number with a negative exponent. The result is not a negative number.

2Try another: Evaluate: 50.04

125

Most calculators can convert from decimal to fraction. Some calculators, you can just press the fraction button then enter. On others, the DF button will convert the decimal to a fraction.

3 21 1So 2 and 5 .

8 25 You may notice, the result is the base to

the positive exponent under one.

nn

If n is apositive integer and b is a real number (b 0), then:

1

bb

.

Definition:

3



Definition: If b is any nonzero real number, b0=1.

Example: 70=1 Try using a scientific calculator to verify this is true. If you don’t have one, then get one. I’m sure your instructor would be more than happy to recommend an appropriate calculator.

More Examples:

015 1 03

111

0n 1 03 4x y 1 , ,,

This definition is consistent with property 5i) from the previous slide.

If we subtract the exponents, the exponent equals zero.

Example:5

5 5 05

xx x 1

x

nn m

mb

5. i) b if n mb

Next Slide

4



Example 1. Evaluate the following:

2a) 4

4b) 3

7c) x

Solutions:

21

a) 4

116

41

b) 3

181

7c)

1

x

Answers:1

a) 32

1b)

49 51

c) x

Your Turn Problem #1

Evaluate the following:

5a) 2

2b) 7

5c) x

5



We’ll now simplify a quotient of two variables with negative exponents. An expression is considered simplified if it does not contain negative exponents and like variables are combined using properties of exponents.

Simplify: 3

5

x

y

(Shortcuts given after example.)

-nn

11. Use the property b = to rewrite without

b negative exponents.

2. Simplify the complex fraction.

3

5

1

x1

y

3 5

1 1

x y

5

3

1 y1x

5

3

y

x

Shortcut: If the exponent is negative, move it from denominator to numerator or numerator to denominator, then change the exponent to positive. If the exponent is positive, leave it where it is.

Next Slide

6



2

7a)

y

x

4 91

b) x

y

7c) x y

Example 2. Simplify the following. Use the shortcut:

7

2x

a) y

4

9x

b) y

7

1x

c) y

Solutions:

Shortcut Procedure: If the exponent is negative, move it from denominator to numerator or numerator to denominator, then change the exponent to positive. If the exponent is positive, leave it where it is.


Simplify the following:12

5x

a) y

3

5x

b) y

9

2x

c) y

5

12y

a) x

3 51

bx

)y

9 2c) x y

Answers:

7



Now that we know how to deal with negative integer exponents, we will be able to simplify more types of problems. Please note: There will usually be more than one method of simplifying these expressions. Recommendations will be given on the “easiest” method, however, it certainly will not be the only method. Recall the properties of exponents and the definition of a negative exponent:

n

n mm

b 5i) b if n m

b

n

m m nb 1

5ii) if n<mb b

nn m

mb

b .b

We can rewrite #5 simply as:

Next Slide

Then use the definition of a negative exponent if necessary:

nn

1b

b

8



5 22

3a) 12 8 x96x

96x n m n mb b b

18 12

2

8c) x y

y

x

n n nab a b

Example 3. Simplify the following:

5 3a) 12x 8x

32b) x

42 3c) x y

Solutions:

66)

1b x

x

mn nmb b

Best to 1st multiply the exponent outside the parentheses with the exponents inside the parentheses. Then use the definition of a negative exponent.


Simplify the following:

7 6a) 7x 9x 22b) x 35c) x y

a) 63x 4

b) 1

x

3

15y

c x

) Answers:

9



5

8 3x

a) x

1

x

n

m m n

Make the exponents positive,b 1

then use if m>n.b b

nn m

m

bOr we could use b first:

b

88 ( ) 3

35

5x

x xx

1

x

Solutions:

Example 4. Simplify the following:8

5x

a) x

These examples involve quotients. We have several options. Some prefer to make the exponents positive first. Others may prefer to use the property:n

n mm

bb .

b

12

10x

b) x

12 20 211

b) x x

1

x

n m n mMake the exponents positive,then use b b b .

nn m

m

bOr we could use b first:

b

1212 10 22

1 20 21x

x xx x

6

1a)

x

192xb)

3

Answers:Your Turn Problem #4


11x

a) x

12

7

12xb)

18x

10



2 5 3a) 4 4 64 n m n mb b b

mn nmb b66

16

1

4b) 2

2

n n nab a b

21 12 1 2 11

3c) 3 2 3

92

22


2 5a) 4 4

Solutions:

23b) 2

12c) 3 2

a) 116

b) 1

125c)

4981

Answers:



5a) 2 2 31b) 5 14 2c) 3 7

11



23 6

2 84

6 8 14

x xa)

xx

x x x

23

4x

a) x

Solutions:


These examples involve quotients with exponents on the outside of the parentheses.

We again have several options. Let’s first use the propertyn n

na a

.b b

35

3x

b) x

35 15 9

3 93 615

x x xb

1)

x x xx

2a) x 12b)

1

x

Answers:



23

4x

a) x

15

7x

b) x

12



2 2

2 2

2 5a

45

25)

2

Example 7. Simplify the following.

22a)

5

Solutions:

More examples of quotients with exponents on the outside of the parentheses. Let’s

first use the propertyn n

na a

.b b

31

23

b) 3

31 3

3 62

3 6 9

3 3b)

33

1 3 3 3 9,683

a) 2516

b) 1

256

Answers:



a) 5

23

12

b) 2

13



631 2 5 3 6 15

3

6

1515x

a) 2 x y 2 x y2 y

x

8y


32 5a) 2x y

Solutions:

It is definitely a good idea to apply the exponent outside the parentheses to all of the exponents inside. Be careful, if a variable doesn’t have an exponent written, then the exponent is 1. The exponent on the coefficients is also a 1. It is a good idea to just write the exponent of 1 to avoid mistakes.

24b) 3xy

821 21 8 2 82 2

8

2y

b) 3 x y 3y

9x y

3 x x

8

4a

y)

25x

9

3y

b) 27x

Answers:Your Turn Problem #8


22 4a) 5x y 33b) 3xy

14



2 3 1

6

4

41x y y

a) 3x

y

x

3


6 3

2 124x y

a) 72x y

Solutions:

Again, there are many methods to simplifying. Just don’t be a rule-breaker and it will work out. Since there are no exponents on the outside of parentheses, reduce the coefficients and make the exponents positive. Then use the appropriate properties. 4 5

240x y

b) 56xy

5 5

2

4 35y

b) 7xx

5

7x yy

6

33x

a 5y

) 4b)

6

7y

Answers:



2 5

8 215x y

a) 25x y

6 3

666x y

b) 77x y

15



31 2 5 3 6 15 3 6

1 4 2 3 12 6 3 12 6 5 11 6 22 x y 2 x y 23 x

a) 3 x y 3 x y 2 x y

7

8x yy


32 5

4 22x y

a) 3x y

Solutions:

It is still a good idea to apply the exponent outside the parentheses to all of the exponents inside. If the coefficients can be reduced (b), do so first to make the operations more manageable.

45 3

210x y

b) 25x y

41 5 3 4 20 12 4 20 12 4

1 2 1 4 8 4

12

4 8

162 x y 2 x y 5 x y 625xyb)

5 x y 5 x y 2

y16x

4 225

1a

6x

y) 9

2b

7

8y)

Answers:



24

1 54xy

a) 5x y

32 5

2 218x y

b) 27x y

16



3 21 1 1 1 9 8

a) 8 9 72 722

17723

1 1 1

3 1

1

1

1 1 1 1 3b)

8 5 402 53

4

4030

4 2 4 2

2 4 2 4

2 4 4 2y x 3y 7x

y

3 7 3 7c)

x y x x yx y

We’re almost done. We just need to cover addition and subtraction of expressions with negative exponents. Make the definition of a negative exponent to make the exponents positive. Then to combine fractions using the LCD.


3 2a) 2 3

Solutions:

13 1b) 2 5

4 2c) 3x 7y

(Do inside parentheses 1st)


Simplify the following: 4 2a) 2 4 13 2b) 2 3 1 1c) 2x 7y

a) 18

b) 7217

2yy

c) 7x

x

Answers:The EndB.R.5-18-07

6.1 using integers as exponents bobsmathclass.com copyright © 2010 all rights reserved. 1 if m and...

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