Warm-up, 3/28 Compute: 1. 2 x 2 x 2 x 2 = 2. 3 x 3 x 3 = 3. 2 x 2 x 3 x 3 x 3 = 4. 5 x 5 x 2 x 2 = 5. 2 x 2 x 4 =
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Warm-up, 3/28
Compute:
1. 2 x 2 x 2 x 2 =
2. 3 x 3 x 3 =
3. 2 x 2 x 3 x 3 x 3 =
4. 5 x 5 x 2 x 2 =
5. 2 x 2 x 4 =
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.
43 = 4 4 4 = 64
Write 64 in standard form.
34 = 3 3 3 3
= 81
What happens if we multiply
23 24?
23 24=
(2 2 2) (2 2 2 2)
23 24
So…23 24= 27 because we used 2 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
Review:
Compute: 25
Answer: 32
Compute: 5-2
Answer: 1/25
Compute: 02
Answer: 0
Compute: 20
Answer: 1
Compute: - 42
Answer: -16
Compute: (-4)2
Answer: 16
Compute: - 4-2
Answer: - 1/16
Compute: 3-2
Answer: 1/9
Compute: (22 )(23)
Answer: 25 or 32
Compute: (33 )2
Answer: 36 or 729
Multiply and express in
exponential form: (45 )(82)
Answer: 216
What is one-third of 399 ?
Answer: 398
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