Power and Scientific Notation
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Power and Scientific Notation
Integer raised to a whole number power
np
np = 1 x (n) p times
usually not important
23 = 1 x 2 x 2 x 2 = 8
1 x (2) 3 times
35 = 3 x 3 x 3 x 3 x 3 = 540
(3) 5 times
Integer raised to a zero power
n0 = 1
np = 1 x (n) p times
usually not important
n0 = 1 x (n) zero times
= 1
20 = 1 x (2) zero times
= 1
5970 = 1 x (597) zero times
= 1
We will revisit this later
Integer raised to the first (1) power
n1 = n
n1 = 1 x (n) one time
= 1 x n
= n
21 = 1 x (2) one time
= 1 x 2
= 2
441 = 1 x (44) one time
= 1 x 44
= 44
Positive Power Worksheet
Power#
0 1 2 3 4 5
1 np
2 20 = 1
3 33
4
5
6 62
7
Power
#
0 1 2 3 4 5
1 1 1 1 1 1 1
2 1 2 4 8 16 32
3 1 3 9 27 81 243
4 1 4 16 64 256 1024
5 1 5 25 125 625 3125
6 1 6 36 216 1296 7776
7 1 7 49 343 2401 16,807
Negative integer raised to a positive power-np
-(np) or (-n)p
very important that you know which one
-(np) : the negative of a positive integer raised to a positive power
-(24) = -(2 x 2 x 2 x 2)
= -(4 x 2 x 2)
= -(8 x 2)
= -(16)
= -16
(-n)p : a negative integer raised to a positive power
(-2)4 = (-2) x (-2) x (-2) x (-2)
= (4) x (-2) x (-2)
= (-8) x (-2)
= 16
Power
#
0 1 2 3 4 5
-1 -10 = 1
-2
-3 -32 =
-3 x -3 =
-4
-5
-6
-7
Power
#
0 1 2 3 4 5
-1 1 -1 1 -1 1 -1
-2 1 -2 4 -8 16 -32
-3 1 -3 9 -27 81 -243
-4 1 -4 16 -64 256 -1024
-5 1 -5 25 -125 625 -3125
-6 1 -6 36 -216 1296 -7776
-7 1 -7 49 -343 2401 -16,807
Positive number raised to a negative power
formula: x-n = 1
xn
2-4 = 1 = 1 = 1
24 2x2x2x2 16
Power
#
0 -1 -2 -3 -4 -5
1
2
3
4 4-2
5
6
7
Power
#
0 -1 -2 -3 -4 -5
1 1 1/1 = 1 1 1 1 1
2 1 1/2 1/4 1/8 1/16 1/32
3 1 1/3 1/9 1/27 1/81 1/243
4 1 1/4 1/16 1/64 1/256 1/1024
5 1 1/5 1/25 1/125 1/625 1/3125
6 1 1/6 1/36 1/216 1/1296 1/7776
7 1 1/7 1/49 1/343 1/2401 1/16,807
Negative number raised to a negative power
formula: (-x)-n = 1
(-x)n
-2-4 = 1 = 1 = 1 -24 (-2) x (-2) x (-2) x (-2) 16
Power
#
0 -1 -2 -3 -4 -5
-1
-2
-3
-4
-5
-6
-7
Power
#
0 -1 -2 -3 -4 -5
-1 1 1/-1 = -1 1 -1 1 -1
-2 1 1/-2 1/4 1/-8 1/16 1/-32
-3 1 1/-3 1/9 1/-27 1/81 1/-243
-4 1 1/-4 1/16 1/-64 1/256 1/-1024
-5 1 1/-5 1/25 1/-125 1/625 1/-3125
-6 1 1/-6 1/36 1/-216 1/1296 1/-7776
-7 1 1/-7 1/49 1/-343 1/2401 1/-16,807
Scientific Notation
• Used to express very large or very small numbers without writing all the zeroes (0)
602200000000000000000000 (6022 with 20 zeroes after it)
0.000000721
Scientific Notation
602200000000000000000000 (6022 with 20 zeroes after it)
100 = 1
101 = 10
102 = 100
103 = 1000
10100 = 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 (googol)
6 x 100 =6
6 x 101 = 60
6 x 102 = 600
6 x 103 = 6000
6 x 1023 = 600000000000000000000000602200000000000000000000
6.022 x 1023 = 602200000000000000000000
Scientific Notation
• Used to express very large or very small numbers without writing all the zeroes (0)
602200000000000000000000 (6022 with 20 zeroes after it)
0.000000721
Scientific Notation
0.000000721
100 = 1
10-1 = 0.1
10-2 = 0.01
10-3 = 0.001
7 x 100 = 7
7 x 10-1 = 0.7
7 x 10-2 = 0.07
7 x 10-3 = 0.007
7 x 10-7 = 0.00000070.000000721
7.21 x 10-7 = 0.000000721
Scientific Notation
Large numbers
Count backward until you get to the start digit of the number
Write a decimal times a power of 10
23
6.022 x 1023
602200000000000000000000 6.022 x 1023
602200000000000000000000
23
Scientific Notation
Small numbers
Count forward until you get to the first number
Write a decimal times a negative power of 10
0.000000721
7
7.21 x 10-7
7.21 x 10-7
0.000000721
7
0.0000000000000000497 0.00000028
Order of Operations
7 + 3 x 5
10 x 5 = 50
Everybody has to agree
} 7 + 3 x 5
7 + 15 = 22}
Order of Operations
• PE[MD][AS]• Parenthesis, Exponents, [Multiplication or Division], [Addition or Subtraction]
left to right• Please Excuse My Dear Aunt Susan
(-4 - 3)3 + 5-3(-4)(-7)3 + 5-3(-4) Parenthesis -343 + 1 (-4) Exponents
125-343 + -4 Multiply/Divide
125-343 4 Add/Subtract
125
PEMDAS
(7+3) x 4 ÷ 2 – 5 x 6 8 + (5)(4) – (6 + 10 ÷ 2) + 42
Power Rules
Power Rule (Powers to Powers)
(an)m = anm
(23)2 = 23*2 = 26 = 64 (23)2 = 23*2 = 26 = 64
(23) x (23) = 8 x 8 = 64 (8)2 = 64
Power Rules
Product Rule
an x am = an+m
23 x 22 = 23+2 = 25 = 32
8 x 4 = 32
Power Rules
Quotient Rule
an
am= an-m
25
23= 25-3 = 22 = 4
328
= 4
NOT a Power Rule
an + am = ?
an - am = ?
No shortcut. Must figure it out the long way.
Same Base
(an)m = anm
an x am = an+m
an
am= an-m
Different Bases
• Do it the long way
23 x 32 = 8 x 9 = 72
25
333227
= = 15
27
Exponent Worksheet
Adding scientific notationPowers of 10 must be the same
• Add the constants
• Keep the power the same
7.21 x 103 + 2.1 x 103
7.21 + 2.1
9.31 x 103
9.21 x 103 + 2.1 x 103
9.21 + 2.1
11.31 x 103
x 1031.131 x 101
1.131 x 101+3 = 1.131 x 104
Subtracting scientific notationPowers of 10 must be the same
• Subtract the constants
• Keep the power the same
7.21 x 103 - 2.1 x 103
7.21 - 2.1
5.11 x 103
9.21 x 103 - 8.31 x 103
9.21 – 8.31
0.9 x 103
x 1039 x 10-1
9 x 10-1+3 = 9 x 102
Adding or subtracting scientific notationPowers of 10 must be the same
3.5 x 104 + 3.1 x 105
104+1? =
5
Make them the same
104 x 101 = 104+1 = 105
3.5 x 101 = .35
.35 x 105 + 3.1 x 105
.35 + 3.1
3.45 x 105
Adding or subtracting scientific notationPowers of 10 must be the same
3.5 x 104 + 3.1 x 105
105-1? =
4
Make them the same
105 x 10-1 = 105-1 = 104
3.1 x 10-1 = 31
3.5 x 104 + 31 x 104
3.5 + 31
34.5 x 104
x 1043.45 x 101
3.45 x 101+4 = 3.45 x 105
Adjusting Scientific Notation
Decimal to the left
10x : biggerDecimal to the right
10x : smaller
3.5 x 104
.35 x 105
3.5 x 104
35.0 x 103
3.5 x 10-4
.35 x 10-3
3.5 x 10-4
35.0 x 10-5
3.5 x 104 + 3.1 x 105
Adjusting Scientific Notation
Decimal to the left
10x : biggerDecimal to the right
10x : smaller
x 105.35 + 3.1 x 105
3.45 x 105
3.5 x 104 + 3.1 x 105
x 10431.03.5 x 104 +
34.5 x 104
3.45 x 105
Multiply scientific notation• Multiply the constants
• Use the multiplication power rule on the powers
(7.21 x 103)(2.1 x 102)
x 103 x 1027.21 x 2.1
15.141 x 103+2
15.141 x 105
(7.21 x 103)(2.1 x 102)
(7210)(210)
1,514,100
1,514,100
Divide scientific notation• Divide the constants
• Use the division power rule on the powers
7.21 x 105
2.1 x 107
105
107
7.21
2.1
x 105-73.43
x
3.43 x 10-2
7.21 x 105
2.1 x 107
721,000
21,000,000
.0343
.0343
Divide scientific notation• How many times larger is 3 x 1010 than 5x 106
3 x 1010
5 x 106
1010
106
3
5
x 1010-60.6
x
0.6 x 104
30,000,000,000
5,000,000
6000
6000
3 x 1010
5 x 106
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