Definition of Let b represent any real number and n represent a positive integer. Then,

n factors of b

nb

...nb b b b b b

Expression Base Exponent Result

6² 6 2 (6)(6) = 36

(-½)³ -½ 3 (½)(½)(½) =

0.8⁴ 0.8 4 (.8)(.8)(.8)(.8) = .4096

1

8

5 is the base to the exponent 4

45

41 5

1 5 5 5 5 625

4( 5)

Multiply -1 times four factors of 5

Parentheses indicate that -5 is the base to the exponent 4.

Multiply four factors of -5= (-5)(-5)(-5)(-5)

= 625

Substitute a = 2

25a Use parentheses to substitute a number for a variable.

= 5 ( )²

2

Simplify

2

= 5(4)

= 20

Substitute a = 2

2(5 )aUse parentheses to substitute a number for a variable.

= [5( )]²

2

Simplify inside the parentheses first

2

= (10)²

= 100

Substitute a = 2 b = -3

25ab Use parentheses to substitute a number for a variable.

= 5( ) ( )²

2

Simplify inside the parentheses first

2

= 5(2)(9)

= 100

-3

-3

multiply

Tip: In the expression 5ab² the exponent, 2 applies only to the variable b. The constant 5 and the variable a both have an implied exponent of 1.

Substitute a = 2 b = -3

2( )a b

= (a + b )²

2

Simplify inside the parentheses first

2

= [(2) + (-3)]²

= (-1)²

-3

-3

= 1

Avoiding Mistakes: Be sure to follow the order of operations. It would be incorrect to square the terms within the parentheses before adding.

Multiplication of Like Bases

Assume that a≠ 0 is a real number and that m and n represent positive integers. Then, Property 1

5 2 7( )( ) ( )x x x x x x x x x x x x x x x x x

m n m na a a

5 2 5 2 7x x x x

Division of Like Bases

Assume that a ≠ 0 is a real number and that m and n represent positive integers such that m > n. Then,

Property 2:

53

2 1

x x x x x x x x xx

x x x

mm n

n

aa

a

55 2 3

2

xx x

x

3 4

3 4 7

w w

w w

(x∙x∙x)(x∙x∙x∙x)

Add the exponents

3 4

3 4 7

2 2

2 2 128 (2∙2∙2∙2∙2∙2∙2)

Add the exponents ( the base is unchanged).

6 2

6

4

4 t

t

t

t

(t∙t∙t∙t∙t∙t) (t∙t∙t∙t)

Subtract the exponents

2

6

4

4

65 5

5

5

5 2

5∙5∙5∙5∙5∙5 5∙5∙5∙5

Subtract the exponents ( the base is unchanged).

6 3 99 3 6

3 3

6 3

3

z z

z

z zz z

z z

Subtract the

exponents

7 77 3 4

2 1

2

3

7

10 1010 10 10,000

10 1

0

0

0

1

10 1

Note that 10 is equivalent to 10¹

Add the exponents in the denominator ( the base is unchanged).

Add the exponents in the numerator ( the base is unchanged).

Subtract the exponents

Simplify

(3p²q⁴)(2pq⁵)

Apply the associative and commutative properties of multiplication to group coefficients and like bases

=(3∙2)(p²p)(q⁴q⁵)

Add the exponents when multiplying like bases.

2 1 4 5(3 2)( )p q

Simplify

3 96p q

Group like coefficients and factors.

Subtract the exponents when dividing like bases.

Simplify

9 3

8

9 3

8

9 8 3 1

2

16

3

16( )( )( )316( )316( )3

w z

w z

w z

w z

w z

wz