Powers of Ten
In exponential notation, the number being raised to a power is called the base and the number
representing the power is called the exponent:
exponent
base
2
10
Study the patterns below to help you understand the positive and negative exponents of base ten:
10 0
10 1
10 2
10 3
10 6
10 9
=
=
=
=
=
=
1
10
100
1000
1,000,000
1,000,000,000
=
=
=
=
10×10
10×10×10
10×10×10×10×10×10
10×10×10×10×10×10×10×10×10
10 -1
10 -2
10 -3
10 -6
10 -9
=
=
=
=
=
0.1
0.01
0.001
0.000001
0.000000001
=
=
=
=
=
1/10
1/10 2
1/10 3
1/10 6
1/10 9
=
=
=
=
=
1/10
1/100
1/1000
1/1,000,000
1/ 1,000,000,000
Scientific Notation
Scientific notation is a shorthand method for expressing very large or very small numbers. In scientific
notation, a single number is rewritten as the product of two numbers, a coefficient and an exponent of
base ten.
For example, in scientific notation the number 365,000,000,000 is written as
coefficient
11
3.65 × 10
In order for a number to correctly written in scientific notation, the following conditions must be true:
1. The coefficient must be greater than or equal to 1 and less than 10.
2. The base must be 10.
3. The exponent must show the number of places that the decimal point must be moved to
write the number in standard notation.
A positive exponent means that the decimal point in the coefficient is moved to the right when changing
to standard notation. In the example above, moving the decimal point 11 places to the right will change
3.65 × 10 11 back to 365,000,000,000 again.
A negative exponent means that the decimal point in the coefficient is moved to the left when changing to
standard notation. For example, the number 3.65 × 10 -11 is written as 0.0000000000365 in standard
notation.
Study the patterns below to help you understand the use of positive and negative exponents in scientific
notation:
3.65
3.65
3.65
3.65
3.65
3.65
3.65
3.65
3.65
×
×
×
×
×
×
×
×
×
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
=
=
=
=
=
=
=
=
=
3.65
3.65
3.65
3.65
3.65
3.65
3.65
3.65
3.65
×
×
×
×
×
×
×
×
×
1/1000
1/100
1/10
1
10
100
1000
10,000
100,000
=
=
=
=
=
=
=
=
=
0.00365
0.0365
0.365
3.65
36.5
365
3650
36500
365000
Multiplying & Dividing
To multiply two numbers in scientific notation, multiply the coefficients and add the exponents:
(A × 10 Y ) × (B × 10 Z) = (A × B) × 10 (Y+Z)
For example:
40,000,000 × 20,000
= (4.0 × 10 7) × (2.0 × 10 4)
= (4.0 × 2.0) × (10 7 × 10 4)
= 8.0 × 10 11
To divide two numbers in scientific notation, divide the coefficients and subtract the exponents:
(A × 10 Y ) ÷ (B × 10 Z) = (A ÷ B) × 10 (Y-Z)
For example:
40,000,000 ÷ 20,000
= (4.0 × 10 7) ÷ (2.0 × 10 4)
= (4.0 ÷ 2.0) × (10 7 ÷ 10 4)
= 2.0 × 10 3
If the coefficient in the result is less than 1 or greater than or equal to 10, you must move the decimal
point and increase or decrease the exponent accordingly. For example:
(6.0 × 10 7) × (2.0 × 10 4) = 12.0 × 10 11 = 1.20 × 10 12
Adding & Subtracting
All exponents must be the same before you can add and subtract numbers in scientific notation. You add
or subtract the coefficients only, not the exponents. If two numbers that you want to add or subtract do
not have the same exponents, then you must move the decimal point of one number until its exponent
equals the exponent of the other number. For example:
(5.55 × 10 9) + (2.75 × 10 7)
If we rewrite (2.75× 10 7 ) as (0.0275 × 10 9 ), we can do the addition:
(5.55 × 10 9) + (0.0275 × 10 9) = (5.55 + 0.0275) × 10 9 = 5.5775 × 10 9
Notice that the result is close to the larger of the two numbers (the one that had the higher exponent). For
this reason, it is usually best to express all the numbers to be added or subtracted in the highest power of
ten that appears in the problem.