Laws of Exponents

Objective:TSW simplify powers.TSW simplify radicals.

TSW develop a vocabulary associated with exponents.

TSW use the laws of exponents to simplify.

Exponents

The lower number is called the base and the upper number is called the exponent.

The exponent tells how many times to multiply the base.

Exponents

73

exponent

base

power

1. Evaluate the following exponential expressions:

A. 42 = 4 x 4 = 16 B. 34 = 3 x 3 x 3 x 3 = 81 C. 23 = D. (-1) =7

Squares

To square a number, just multiply it by itself.

                  

   

=

= 3 x 3 = 93 squared =

Perfect Squares

1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49

8² = 64 9² = 81 10² = 100 11² = 121 12² = 144 13² = 169

Square Roots

A square root goes the other direction. 3 squared is 9, so the square root of 9 is 3

3 9

Square Roots

11

4 2

9 3

16 4

25 5

36 6

749

864

981

10100

11121

12144

13691

Radicals

- The inverse operation of raising a number to a power.

For Example, if we use 2 as a factor with a power of 4, then we get 16. We can reverse this by finding the fourth root of 16 which is 2.

= 2164

Radicals

In this problem, the 16 is called the radicand, the 4 is the index, and the 2 is the root.

The symbol is known as the radical sign. If the index is not written, then it is understood to be 2.

The entire expression is known as a radical expression or just a radical.

Example

Simplify:

a) c)

b) d)

8127

16 8

34

3

Laws of Exponents

Whenever we have variables which contain exponents and have equal bases, we can do certain mathematical operations to them.

Those operations are called the “Laws of Exponents.”

Laws of Exponents

m

mm

mnnm

mmmnmnm

y

x

y

xxx

yxxyxxx

.4.3

.2.1

Laws of Exponents

mnn

m

nmn

m

xx

xthenmnifb

xx

xthennmifa

1,.5

,.5

Zero Exponents

A nonzero based raise to a zero exponent is equal to one

a0 = 1

Negative Exponents

a-n= (

1______

an )

A nonzero base raised to a negative exponent is the reciprocal of the base raised to the positive exponent.

A nonzero base raised to a negative exponent is the reciprocal of the base raised to the positive exponent.

Basic Examples

32 xx 32x 5x

34x 34x 12x

Basic Examples

3xy 33yx

3

y

x3

3

y

x

4

7

x

x

1

47x 3x

7

5

x

x 57

1

x 2

1

x

Basic Examples

Examples

5

p

u

74 xx

57y

3

9

x

x

1.

2.

3.

4.