Exponents

Review:

•What do exponents tell us?

•What does 53 mean?

53 =5 5 5

What is the “base”?What is the “exponent”?

8 5Base

Expone

nt

Write 63 in standard form.

63 = 6 6 6 = 216

Write 64 in standard form.

64 = 6 6 6 6

= 1,296

What happens if we multiply

63 64?

63 64=

(6 6 6) (6 6 6 6)

63 64

So…63 64= 67 because we used 6 as a factor seven times!

Notice anything?

Let’s try another example:

What is 52 54?

52 54 = (5 5) (5 5 5 5)or

52 54 = 56

For MULTIPLICATION:

x s x t = x s + t

(For any number x and for integers s and t.)

• In other words, when multiplying two numbers in exponential form with the same base, add the exponents.

Ex:9 4 9 8 = ?

9 12

What happens if we divide?75

73

75 = 7 7 7 7 7

73 7 7 7

Cancel out any numbers that are in the numerator AND denominator!

After we cancel, we are left with 72.

So, 75 = 72

73

For DIVISION:

xs =

xt

(For any nonzero number x and for integers s and t.)

x s - t

• In other words, when dividing two numbers in exponential form with the same base, subtract the exponents.

Ex:6 8 ÷ 6 5 = ?

6 3

What about (52)3?(52)3

(52) (52) (52)

(5 5)(5 5)(5 5)

56

Raising a power to a power:

(xs)t = x s t

(For any number x and for integers s and t.)

• In other words, when raising a number with an exponent to a power, multiply the exponents.

Ex:

(5 3) 7 = ?

521

One important thing to remember…

x0 = 1

(Any number to the zero power is equal to 1!)

Why???

•For example…

32 ÷ 32 = 3 2 - 2 = 30

32 = 3 3 = 9 = 1

32 3 3 9

•So, 30 = 1

•This works for any base number!

Can we have negative exponents?

• Yes! Negative exponents represent the reciprocal.

• The reciprocal of x is

The reciprocal of 10 is

• They do NOT make the answer negative.

x

1

10

1

5 -2 = =

2-4=

25

1

25

1

16

1

2

14