1

Summer Assignment Review

Exponent Rules

2

Properties of ExponentsProduct of Powers PropertyLet a be a real number, and let m and n be positive integers.

nmnm aaa To multiply powers having the same base, add the exponents.

**Each individual property is simple. The difficulty is keeping them straight once you have learned them all, and in remembering that the operation you perform on the exponents is not the operation that is occurring between the powers.

Examples114747 9999

75252 xxxx

15312 aaa

3

Properties of ExponentsTo multiply monomials, re-order the factors in order to multiply powers with like bases.

79375425742 15))()()(53()5)(3( cbacbbaaabcba

I call this a training wheel step. This work can be done in your head, or by drawing arrows between the like bases on the original problem.

Examples

Properties of Exponents

4

Power of a Power PropertyLet a be a real number, and let m and n be positive integers.

nmnm aa To raise a power to a power, multiply the exponents.

Examples This property is often confused with the product of powers property. Because of this you need to pay close attention to these two. Look at how they are related.

8422222222242 xxxxxxxx

2483

459595

84242

777

aa

xxx

5

Properties of ExponentsPower of a Product PropertyLet a be a real number, and let m and n be positive integers.

To find a power of a product, find the power of each factor.

Examples

mmm baab )(

3333)( zyxxyz 555 32)2( kjjk

When applying all three of these rules there are multiple paths to a simplified expression.

Be careful, don’t multiply the constant by the power!

6

Properties of ExponentsPractice

1)2)3)4)5)6)

7)8)9)

10)11)

7

Properties of ExponentsQuotient of Powers PropertyLet a be a nonzero real number, and let m and n be positive integers such that m > n

0, aaa

a nmn

mTo divide powers having the same base, subtract the exponents

We are actually cancelling out matching factor pairs…

23

5

1x

xx

xxx

xxxxx

x

x

Subtracting accomplishes the same goal with out all that work.

2353

5

xxx

x

8

Properties of ExponentsPower of a Quotient PropertyLet a and b be a real numbers with b≠0, and let m be a positive integer.

To find a power of a quotient, find the power of the numerator and the power of the denominator and divide.

Basic Examples:

0,

bb

a

b

am

mm

4

44

y

x

y

x

22

224977

aaa

Using more than one property:

3

6

33

32332

125

64

5

4

5

4

y

x

y

x

y

x

Training wheels step

9

Properties of ExponentsPractice

1) 2)3) 7) 8)9)

4) 5)6)

10

Properties of Exponents

Zero Exponents Negative Exponents

0,10 aaa to the power of zero is 1.

1137

450

154638

1

15

0

5

79

0

0

0

z

yx

x

0,1

aa

an

n

0,1

aa

an

n

22 1

xx

A simplified expression can not contain any negative exponents

44

4 1

1x

x

x

11

Properties of ExponentsPractice