Lesson 5.1

Exponents

Definition of Power:

If b and n are counting numbers, except that b and n are not both zero,

then exponentiation assigns to b and n a unique counting number bn, called a power.

bn n is called the exponent.b is called the base.

How do we compute exponents?

The exponent counts the number of times that 1 is multiplied by the base.

OR

The exponent n counts the number of times that the expansion · b is joined to 1.

Power Form

Number ofExpansions

Operator Model

BasicNumeral

34 1 · 3 · 3 · 3 · 3 81four

33 1 · 3 · 3 · 3 27three

32 1 · 3 · 3 9two

31 1 · 3 3one

30 1 1none

In general, bn = 1 · b · b · b · … · b

n times

For all b 0, b0 = 1.

Here are several more examples:

33 = 1 · 3 · 3 · 3 = 27 32 = 1 · 3 · 3 = 9 31 = 1 · 3 = 3 30 = 1

23 = 1 · 2 · 2 · 2 = 8 22 = 1 · 2 · 2 = 4 21 = 1 · 2 = 2 20 = 1

13 = 1 · 1 · 1 · 1 = 1 12 = 1 · 1 · 1 = 1 11 = 1 · 1 = 1 10 = 1

03 = 1 · 0 · 0 · 0 = 0 02 = 1 · 0 · 0 = 0 01 = 1 · 0 = 0

We have the following general rules:

For all b, b1 = b. For all n 0, 0n = 0.

? ?

What about 00 ? It is not defined.

Evaluate each power:

By definition, 62 = 1 · 6 · 6 = 36

When the exponent is greater than 1, this can be shortened by dropping the 1 at the front and writing: 62 = 6 · 6 = 36.

62

24 We may compute this as 24 = 2 · 2 · 2 · 2 = 16

50 Since the exponent is 0, we have to use the definition.50 is 1 multiplied by 5 no times. That is, 50 = 1.

We may use exponents to count the number of factors:

7 · 7 · 7 · 7 may be written as 74.

b · b · b may be written as b3.

5 · 3 · 3 · 5 · 3 · 3 may be written as 34 · 52.

Caution: You may have been told that 23 is 2 times itself 3 times.

This is NOT TRUE !!

We may write 23 = 2 · 2 · 2,

but there are only 2 products here, not 3.

Remember: For 23, the exponent 3 counts the number of times

that 1 is multiplied by 2.

23 = 1 · 2 · 2 · 2