Math 101 Intermediate Algebra    Exponents and Scientific Notation
Chapter 5, Sections 1 and 2

Some basics about exponents can be reviewed in Chapter 1, section 4.

Rules of Exponents

Rule Example
x0 = 1 60 = 1
xrxs = xr + s 2324 = 23 + 4 = 27
 xrxs =  xr - s,  x 0
 6463 = 64 - 3 = 61 = 6
(xr)s = xrs (23)4 = 23·4 = 212
(xy)r  =  xryr (2·6)2 = 2262 = 4·36 = 144
 ( xy ) r = xryr ,  y 0
 ( 62 ) 2 = 6222 = 36 4 =  9
 x-r  = 1 xr ,  x 0
 2-3  = 1 23 = 18
 ( xy ) -r = ( yx ) r ,  x,y 0
 ( 26 ) -2 = ( 62 ) 2 =  9

Scientific Notation

Scientific Notation is used to express decimal numbers in a form such that there is a number with one non-zero digit to the left of the decimal point multiplied by an appropriate power of 10.

In Scientific Notation Not in Scientific Notation
3.426 × 106 3426000.0
3.426 × 10-6 0.000003426

Learn Rules by Example

Example 1:  Scientific notation to decimal

3.1415 × 107 = 3.1415 × 10000000 = 31415000.0

• The exponent (7) is positive.
• The decimal point is moved to the right 7 places.
• Exponent 1 means number 10.

Example 2:  Scientific notation to decimal

3.1415 × 10-7 = 3.1415 ÷ 10000000 = 0.00000031415

• The exponent (-7) is negative.
• The decimal point is moved to the left 7 places.
• Exponent -1 means number is in the interval (0, 1).

Example 3:  Decimal to scientific notation

123456789.0 = 1.23456789 × 100000000 = 1.23456789 × 108

• The decimal point had to be moved to the left 8 places until there was only one non-zero digit to the left of the decimal point.
• The exponent (8) is positive.
• Number 10 means exponent is 1.

Example 4:  Decimal to scientific notation

0.0000123456 = 1.23456789 ÷ 100000 = 1.23456789 × 10-5

• The decimal point had to be moved to the right 5 places until there was only one non-zero digit to the left of the decimal point.
• The exponent (-5) is negative.
• Number in the interval (0, 1) means exponent -1.

Example 5:  Decimal numbers in the interval [1, 10)

2.34 = 2.34 × 100 = 2.34

• 2.34 is already in scientific notation

Rules in Words

Scientific Notation to Decimal

1. Observe the exponent on the base 10.
2.  (a) If the exponent is positive, move the decimal point to the right the same number of places as the exponent. You may have to add zeros to the number. This will result in a number 10. (b) If the exponent is 0, do not move the decimal point. Drop the factor 100. This will result in a number in the interval [1, 10). (c) If the exponent is negative, move the decimal point to the left the same number of places as the absolute value of the exponent. You may have to add zeros to the number. This will result in a number in the interval (0, 1).

Decimal to Scientific Notation

1. Move the decimal point in the number to the right of the first non-zero digit.
• This results in a number in the interval [1, 10).
2. Count the number of places you moved the decimal point in step 1.
• If the original decimal number is 10 or greater, the count is positive.
• If the original decimal number is in the interval (0, 1), the count is negative.
3. Multiply the number obtained in step 1 by 10 raised to the count (i.e., power) found in step 2.

Using Scientific Notation

Idea
1. Write given numbers in scientific notation
2. Use rules of exponents on the powers of 10.

Multiplication

 (2000000)(40000) =   (2 × 106)(4 × 104) =   (2)(4) × (106)(104) =   8 × 106 + 4 =   8 × 1010 =   80000000000

Division

40000
0.002
=   4 × 104
2 × 10-3
=
 42 × 10410-3
=   2 × 104 - (-3)
=   2 × 107
=   20000000