simplifying exponential expressions. exponential notation base exponent base raised to an exponent...

Simplifying Exponential Expressions

Exponential Notation

BaseExponent

Base raised to an exponent

Example: What is the base and exponent of the following expression?

272 is the

exponent7 is the base

Goal

To write simplified statements that contain distinct bases, one whole number in the numerator and one in the denominator, and no negative exponents.

Ex:

21 4 3 8 4

2 122 1 2

9 9

46

a b b c

aa b c

Multiplying TermsWhen we are multiplying terms, it is easiest

to break the problem down into steps. First multiply the number parts of all the terms together. Then multiply the variable parts together.

Examples:

a. ( 4x )( -5x ) = ( 4 . -5 )( x . x )

= -20x2

b. (5z2)(3z)(4y)= (5.2.3.4)(y.z.z)

= 120yz2

Only the z is squared

Exploration

Evaluate the following without a calculator:

34 =

33 =

32 =

31 =

Describe a pattern and find the answer for:

30 =

81

27

9

3

1

÷ 3

÷ 3

÷ 3

÷ 3

Zero Power

Anything to the zero

power is oneCan “a” equal zero?

a0 = 1

No.

You can’t divide by 0.

xxxxxxx

7x

3 4x

Exploration

Simplify: x x3 4Use the

definition of exponents to

expand

There are 7 “x” variables

Notice (from the initial

expression) 3+4 is 7!

Product of a Power

If you multiply powers having the same base, add the exponents.

m na

Example

Simplify:

32 9x

2 9 03x x y

113x

1

Add the exponents

since the bases are the same

Anything raised to the 0 power is 1

Practice

Simplify the following expressions:

5

3 0 2

3 5 2 4

1)

2) 2 3

3) 9 4

x x

x z x

x y x y

6x56x

5 936x y

3 3 3 3 3x x x x x

15x

3 5x

Exploration

Simplify: 53x

Use the definition of

exponents to expand

The Product of a Power

Rule says to add all the 3s

Adding 3 five times is equivalent to multiplying

3 by 5. The same exponents from the initial

expression!

Power of a Power

To find a power of a power, multiply the exponents.

m na

1 12 9 22 4s s t t

Example

Simplify: 6 32 3 22 4s s t t

13 118s t

2s 2 6s 3 3t 24t

2 4 1 12s 9 2t

Multiply the powers of a

exponent raised to another

power

Any base without a power, is

assumed to have an

exponent of 1

Multiply numbers without

exponents and add the

exponents when the bases

are the same

Practice

Simplify the following expressions:

42

2 54 3

2 65 4 2

1)

2)

3)

y

a a

x y x x y

8y23a

17 11x y

5x

2 2 2 2 2z x z x z x z x z x

5 10x z

Exploration

Simplify: 52z x

2 5z Use the definition of

exponents to expand

The Product of a Power Rule

says to add the exponents with the same bases

Adding 2 five times is equivalent to

multiplying 2 by 5

Notice: Both the z2 and x were raised to the 5th power!

Power of a Product

If a base has a product, raise each factor to the power

m ma b

20y

20y

2x 4 5y 23

Example

Simplify: 52 43 2x xy

7 20288x y

52 5x

2x9 32 5x2 5x 9 32

Everything inside the

parentheses is raised to the

exponent outside the

parentheses

Multiply numbers without

exponents and add the

exponents when the bases

are the same

Multiply the powers of a

exponent raised to another power

Practice

Simplify the following expressions:

5

4 52 3

32 4

1)

2) 2 2

3) - 2 3

pqr

ab a

x x yz

5 5 5p q r

19 8512a b

7 3 1254x y z

First Four

1. 125x3 9. -15a5b5 16. 64x11

2. 64d6 10. r8s12 17. 256x12

3. a7b7c 11. 36z11 18. 9a8

4. 64m6n6 12. 18x5 19. 729z10

5. 100x2y2 13. 4x9 20. 321

6. -r5s5t5 14. a4b4c6 21. 108a11

7. 27b4 15. 125y1222. -81x17

8. -4x7

Exploration

55 312554 62553 12552 2551 550 15-1

5-2

5-3

5-4

1/32 1/16 1/8 ¼ ½ 1

1

25

1

24

1

23

1

22

1

21

1

20

1

2 1

1

2 2

1

2 3

1

2 4

1/5

1/251/1251/625

2

4

8

16

Complete the tables

(with fractions) by finding

the pattern.

÷ 5

÷ 5

÷ 5

÷ 5

÷ 5

÷ 5

÷ 5

÷ 5

÷ 5

x 2

x 2

x 2

x 2

x 2

x 2

x 2

x 2

x 2

Negative Powers

A simplified expression has no negative exponents.

1ma

ma1ma

Negative Exponents “flip”

and become positive

3b

3

620ba

6 320a b

Example

Simplify: 10 3 44 5a b a

4 5 10 4a All of the old

rules still apply for negative exponents This is not

simplified since there is a negative exponent

Flip ONLY the thing with the

negative exponent to the bottom and

the exponent becomes positive

12

2 3128xy

Example

Simplify:2 1

3

12

8

x y

x

532xy

4

4

2x8

3x1y

Everything with a positive

exponent stays where it is.

Since all of the negative

exponents are gone, apply all of the old rules

to simplify.

Everything with a negative exponent

is flipped and exponent becomes

positive.

Practice

Simplify the following expressions:

3

2

5 3

3 8 4

23

1) 8

62)

4

3) 3 y

4) 2

x

x y

x x

a b

1512

7

332xy

8

7

3y

x

6

24ab

x x x x x x x x x xx x x x x x

10 6x

Exploration

Simplify: 10

6xx

4x

Use the definition of

exponents to expand

Since everything is

multiplied, you can cancel

common factors

The 6 “x”s in the denominator cancel 6

out of the 10 “x”s in the numerator. This is the same as subtracting

the exponents from the initial expression!

Only 4 “x”s remain in the

numerator

Quotient of a Power

To find a quotient of a power, subtract the denominator’s

exponent from the numerator’s exponent if the

bases are the same.

a0

m na

6 2x 26

4 213 x y

Example

Simplify:6

2 3

2

6

x y

x y

4

23xy

1 3y

1

Divide the base

numbers first

Subtract the exponents of

the similar bases since

there is division

Not simplified since there is a

negative exponents

Flip any negative

exponents

Practice

Simplify the following expressions:6 0

3

6

12

9 3

3

1) 5

122)

4

143)

4

a b

a

x

xy

x y

x y

3

5a

5

123xy

6 272x y

a a a a a ab b b b b b a a a a a ab b b b b b

Exploration

Simplify:6

a

b

6

6ab

Use the definition of

exponents to expand

Multiply the fractions

Use the definition of exponents to rewrite.

Notice: Both the numerator and

denominator were raised to the 6th power!

To find a power of a quotient, raise the denominator and numerator to the same power.

Power of a Quotient

m

m

a

b

2y 5 3x 2 6 21

2 15

8

3

y x y

x

Example

Simplify:32 2 7

5

3 2x y

y x

2 6 21

15

8

9

y x y

x

2 3x 7 3y 3223

2 21

15 6

8

9

y

x

23

9

8

9

y

x

Everything in the fraction is raised to the

power out side the parentheses.

Multiply the fractions

Subtract the exponents

when there is division, and

add when there is multiplication

Practice

Simplify the following expressions:

2

83

0

4

2

75

4

1)

22)

3)

a

bc

x

y

s f

zr

24

8ab

8

416

y

x

35

28 14 7

f

r s z