B11B11Exponents and Exponents and

Scientific NotationScientific Notation

Multiplying Powers with Like Bases

For any rational number a, and for all whole numbers m and n,

m n m na a a

Example 1Simplify. Express using exponents.a) 3 62 2b) 2 4a ac) 5 3z z z

d) 2 3 4 5(x y )(x y )

926a

9z6 8x y

Practice

a) 2 45 5b) 5 3a ac) 2 3 4y y y

d) 2 2 5(mn )(mn )

Simplify. Express using exponents.

Dividing Powers with Like Bases

For any rational number a except 0, and for all whole numbers m and n,

mm n

na aa

Example 25

377

Simplify. Express using exponents.

a)7

3yyb)4 3

2r srsc)

27

4y

3r s

Practice

a)

b)

c)

Simplify. Express using exponents.7

355

3

2hh8 4

3x yxy

Negative Exponents

For any rational number a except 0, and for all whole numbers m and n,

mm1a a

Example 3Express using positive exponents.

16a)

25b)

4xc)

4xyd)

16

215

41x

4xy

Practice

1)

2)

3)

Express using exponents.22

4y

23c

The Exponent Zero

a0 = 1 for any rational number a except 0.

0a 1

Example 4Simplify.

24a)

31b)

04c)

0xd)

214

311

1

1

116

1

Practice

1)

2)

3)

Simplify.32

91

03

PracticeWrite without exponents.1)

322)

12

3)

25

4)

3 4 27 7 7

Simplify. Express using exponents.

5)

4 4 4 45 5 5 5

Raising a Power to a Power

For any rational number a, and any whole numbers m and n,

m n mn(a ) a

Example 1Simplify. Express using exponents.

(52)3 = 56

(45)6 = 430

(x4)7 = x28

Practice

1) (54)3

Simplify. Express using exponents.

2) (22)5 3) (a6)3 4) (n4)4

Example 2Simplify.

(5x)3 = (5x)(5x)(5x)(3z)2 = (3z)(3z)

(2y2)4

= (2y2)(2y2)(2y2)(2y2)

= 125x3

= 9z2

= 16y8

Practice

1) (3y)2

Simplify.

2) (6m)4

3) (2a3)3 4) (4x3)2

Example 3Simplify.

(4x5y2)3 = 43x15y6

(-2x5y2)7 = -27x35y14

(2y2)4

= (2y2)(2y2)(2y2)(2y2)

= 64x15y6

= -128x35y14

= 16y8

Practice

1) (4y3)4

Simplify.

2) (3x4y7z6)5

3) (-7x9y6)2

Example 4Simplify.

23x5

6

2x5

6x25

45

3xy

20

12xy

PracticeSimplify1) (34)3

33

5xy

2) (6x)3 3) (3x5)4

4) (-3m4n2)2

5)

Multiplying and Dividing Multiplying and Dividing MonomialsMonomials

Example 1Multiply.

(7y)(2y)

= (7)(2)(y)(y)

= 14y2

(5a3)(3a2) = (5)(3)(a3)(a2)= 15a5

(-3x3)(4xy5) = (-3)(4)(x3)(x)(y5)

= -12x4y5

Practice

1) (3x)(-5)

Multiply.

2) (-m)(m)

3) (-x)2x3 4) (3p5q2)(4p2q3)

PracticeMultiply.

5) (4x5y5)(-2x6y4) 6) (-7y4)(-y)(2y3)

7) (7a5)(3a3)(-a5) 8) (9b2)(2b5)(-3b7)

Example 2Divide.

7

4yy

7 4y 3y

9

46a8a

2 5

4 39a b3a b

=34

a5

=3a

b2

2

Practice

1)

Divide.8

5xx 2

)

5

812m8m

3)

3 4

25x y5x y

4

)

15 7

14 632x y8x y