R.2 Integer Exponents

Definition

(i) for any real number a and a positive integer n (1,2,3,4….) a n = a ⋅ a24a

1 ⋅ K3

4 ⋅ n times

3

Example: (2) = 2⋅2⋅2= 8

(ii) for any nonzero real number a: a0 = 1

Example: (-312)0= 1

1

an

Remark: a in the above definition is called the base of the exponent and n is called an exponent or a power

Example: ( −3) −2 = 1 2 = 1

( −3)

9

(iii) for any nonzero real number a and a positive integer n: a −n =

Caution: -32 = -9 but (-3)2 = 9

a n ⋅ a m = a n+m an 1

= a n− m = m− n m a a n m n⋅m (a ) = a

Laws of exponents (must be memorized)

1

2 −5 ⋅ 2 4 = 2(−5 )+4 = 2 −1 =

2

7

2

= 27−5 = 2 2

25

((−3) )

4 2

= ( −3) 4⋅2 = (−3)8

( 2 x) 5 = 25 ⋅ x 5

( a ⋅ b) n = a n ⋅ b n

2

n

22

4

2

= 2 = 2 x x

x

an

a

= n b b

Note that these properties can also be applied in reverse

x 5 = x 2+3 = x 2 ⋅ x 3

( )

x 9 = x 3⋅3 = x 3

3

81x 4 = 34 ⋅ x 4 = (3x) 4

Additional properties of exponents:

−n

b

a

=

a

b m −n a b

= n

−m

b a n

−2

2

32 9

3

2

= = 2 =

2

4

2

3 x −3 y 5

=

y −5 x 3

Caution: Identify the base of the exponent properly. Example:

1

1

x2

=

=

2 x −2 2 ⋅ ( x) −2 2

−2

−1

Example: Simplify the following expression 4 x ( yz )

23 x 4 y

4 x −2 ( yz ) −1

4

41

1

/

= 6 2

= 2 4

= 2 2+4

3 4

1

2 x y x 8 x y ( yz )

8 x yyz 2 x y z

/

R.8 n-th Radicals; Rational Exponents

A square root of a nonnegative number a is a number b such that b2 = a.

Example : square roots of 25 are 5 and -5 since 52 = 25 and (-5)2 = 25

Square root of 0 is 0, since 02 = 0

Square root of – 4 does not exist (in the real number system), since there is no real number that squared gives (-4)

The principal square root, or radical number a is called a radicand.

25 = 5 ,

Example

0 = 0,

, of a nonnegative number a is a nonnegative number b such that b2 = a. The

− 4 not defined

Remark : a) the principal square root of a is often called the square root of a or radical of a

b) the square root of a is NEVER negative

Properties of radicals

( a)

2

( 3)

2

= a,

a≥0

( −5) 2 =| −5 |= 5

a 2 =| a | a ⋅ b = a ⋅ b, a =3

a

=

b

b

am =

( a) ,

,

9⋅5 = 9 ⋅ 5 = 3 5

a, b ≥ 0

4

=

25

m

a≥0

25

16 3 =

b≠0

4

( 16 )

=

3

2

5

= 4 3 = 64

To simplify a radical means to remove all factors that are perfect squares

Example : 12 x 5 = 3 ⋅ 4 ⋅ x 4 ⋅ x =

Note that

x8 =

(x )

4 2

( )

4 ⋅ x2

2

⋅ 3x = 4

= x 4 and, in general,

x 5 = x 4 ⋅ x = x 2 x and, in general,

(x )

2 2

⋅ 3x = 2 x 2 3x

x even = x even / 2 x odd = x ( odd −1) / 2 x

If two expressions contain the same radical (same index and same radicand), then that radical can be factored out and the two expressions combined.

Example: 3 12 − 4 27 = 3 4 ⋅ 3 − 4 9 ⋅ 3 = 3 ⋅ 2 3 − 4 ⋅ 3 3 = 6 3 − 12 3 = 3 (6 − 12) = −6 3

To rationalize the denominator (or numerator) is to eliminate the radical from the denominator (or numerator) through some algebraic operations

Denominator is

Multiply the numerator and the denominator by

a x

The denominator becomes a x ⋅ x = ax

x

(a − b x )(a + b x ) = a − (b x )

a +b x

a −b x

a −b x a u +b w

a +b x

a u −b w

2

a u +b w

2

(a

a u −b w

)(

= a2 − b2 x

) ( ) (

2

)

2

u − b w a u + b w = a u − b w = a 2u − b 2 w

Example: Rationalize the denominator

a) − 3 = − 3 ⋅ 5 = − 15 = − 15

2⋅5

10

2 5 2 5⋅ 5

b) 2 − 3 = 2 − 3 ⋅ 1 + 5 = 2 + 2 5 − 3 − 3 5 = 2 + 2 5 − 3 − 15

2

−4

1− 5

1− 5 ⋅ 1+ 5

12 − 5

(

(

)(

)(

)

)

( )

Higher order radicals and rational exponents

If n > 2 then

-if n is an odd number, then the n-th radical of a, denoted

n

a , is such a number b that bn = a.

-if n is an even number, then the n-th radical of a