power and scientific notation - sph ms....

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