EXPONENTS and ROOTS

Positive Exponents:If a is a real number and n is a positive integer, then an is the product of n factors,each of which is a.

an = a a a . . . a| {z }n factors of a

a is called the base, n is the exponent or power, and an is an exponential expression.

Examples:

(a) 53 = 5 5 5 = 125 (b) (2)4 = (2) (2) (2) (2) = 16

Negative Exponents:an =

1

an, a 6= 0

Examples:

(a) 42 =1

42=

1

16(b) (5)1 =

1

5

Radicals:npa = b means a = bn

Examples: a) 2p25 = 5 because 52 = 25

b) 3p8 = 2 because (2)3 = 8

When n = 2 we say square root and use the symbolp

. When n = 3 we say

cube root and use the symbol3p

.

In general, if n 2 is an integer and a is a real number, we havenpan = a if n 3 is odd

npan = |a | if n 2 is even

1

Radicals are used to define Rational ExponentsIf m and n are integers containing no common factors, with n an integer such thatn 2 and a is a real number, we have

npa = a

1n

amn = n

pam

Examples:

3p8 = 8

13 = 2

1634 =

4p163

Rules for Exponents:If m and n are rational numbers, and a, b and c are real number for which theexpressions below exist, then

am an = am+n product rule(am)n = amn power of power rule

(ab)n = anbn power of a product rulea

c

n=

an

cn, c 6= 0 power of a quotient rule

am

an= amn, a 6= 0 quotient of powers rule

a0 = 1, a 6= 0 zero exponent

Properties of Radicals:npab = n

panpb

n

ra

b=

npa

npb

npam =

npam

Examples:

(a)

r9

4=

p9p4=

3

2

(b) 1634 =

4p163

= (2)3 = 8

2

1. Evaluate the following expressions:

(a) (3)3 =

(b) 53 =

(c) 21 =

(d) 32 =

(e) (36)1/2 =

(f)

3

2

2=

(g)

3

2

1=

(h) 274

3 =

(i) 82/3 =

3

2. Simplify the following expressions:

(a) x3 x2 =

(b)x5

x2=

(c) x3 x2 x1 =

(d)a7

a2=

(d) (3a)2 =

(e)

3xy

2a

2=

(f)

3a

2

2=

4