powers and exponents 53535353 base the base is the number that is multiplied. exponent the exponent...

Powers and ExponentsPowers and Exponents

5533

The basebase is the number that is multiplied.

The exponentexponent tells how many times the base is used as a factor.

The number that is expressed using an exponent is called a powerpower. “Five to the Third Power”

5533 = 5×5×5=125= 5×5×5=125

Exponential NotationExponential Notation““five to the third power”five to the third power”

Expanded NotationExpanded NotationStandard NotationStandard Notation

105 = 1 x 10 x 10 x 10 x 10 x 10 = 104 = 1 x 10 x 10 x 10 x 10 = 103 = 1 x 10 x 10 x 10 = 102 = 1 x 10 x 10 = 101 = 1 x 10 = 100 = 1 =

= 1 x = = 1 x x = = 1 x x x =

Place Value and Powers of TenPlace Value and Powers of Ten ____ ____ ____ , ____ ____ ____. ____ ____ ____

100,000100,000101055 101044 101033 101022 101011 101000

1010--11 1010--22 1010--33

10,00010,0001,0001,000100100101011

10-11

10 0.10.110-2 1

101

10 0.010.0110-3

110

110

110 0.0010.001

Any number to the power of zero is 1.Any number to the power of zero is 1.

Remember…Remember…

121200 = 1 = 1

565600 = 1 = 1

23423400 = 1 = 1

You try…You try…

1) 73

Express each number in both expanded and standard notation.

343343

2) 54 625625

3) 26 6464

4) 240 11

5) 35 243243

1 x 7 x 7 x 71 x 7 x 7 x 7

1 x 5 x 5 x 5 x 51 x 5 x 5 x 5 x 5

1 x 2 x 2 x 2 x 2 x 2 x 21 x 2 x 2 x 2 x 2 x 2 x 2

1 1

1 x 3 x 3 x 3 x 3 x 3 1 x 3 x 3 x 3 x 3 x 3

Scientific Notation Scientific Notation is is a shorthand way of a shorthand way of expressing very large expressing very large or very small or very small numbers.numbers.

An ordinary quarter contains about An ordinary quarter contains about 97,700,000,000,000,000,000,000 97,700,000,000,000,000,000,000 atoms. atoms.

The length of this number in The length of this number in standard standard notationnotation makes it awkward to work with. makes it awkward to work with.

To express any number in To express any number in scientific scientific notationnotation, write it as the product of a , write it as the product of a power of ten and a number greater than power of ten and a number greater than or equal to 1 but less than 10.or equal to 1 but less than 10.

97,700,000,000,000,000,000,00097,700,000,000,000,000,000,000..

9.779.77

11 22 44 55 66 77 88 99 1100

1111

1122

1133

1144

1155

1166

1177

1188

1199

33

x 10x 102222

2200

2211

2222

..

The average size of an atom is about The average size of an atom is about 0.00000003 centimeters across.0.00000003 centimeters across.

Again, the length of this number in Again, the length of this number in standard notationstandard notation makes it awkward to makes it awkward to work with.work with.

To express any number in To express any number in scientific scientific notationnotation, write it as the product of a , write it as the product of a power of ten and a number greater than power of ten and a number greater than or equal to 1 but less than 10.or equal to 1 but less than 10.

0.000000030.00000003..11 22 44 55 66 77 8833

33 x 10x 10--88

..

A A positivepositive exponent means move the exponent means move the decimal to the decimal to the right, right, and a and a negativenegative exponent means move the decimal to exponent means move the decimal to the the left.left.

Helpful HintHelpful Hint

You try…You try…

1) 135,000

Express each number in scientific notation.

1.35 x 101.35 x 1055

2) 0.00001653 1.653 x 101.653 x 10--55

3) 4,600,000 4.6 x 104.6 x 1066

4) 0.007089 7.089 x 107.089 x 10--33

5) 0.000000021 2.1 x 102.1 x 10--88

Let’s try in reverse…Let’s try in reverse…

1) 7.02 x 104

Express each number in standard form.

70,20070,200

2) 9.12 x 10-2 0.09120.0912

3) 6.4 x 107 64,000,00064,000,000

4) 8.8 x 10-5 0.0000880.000088

5) 3.49 x 109 3,490,000,0003,490,000,000