# Scientific Notation

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Scientific Notation Handout
Scientific notation is based on powers of ten. Numbers that consist of a one
followed by several zeros can be written as powers of ten. The power of ten is just the
number of zeros in the number.

10 = 101                                   0.1 = 10-1
100 = 102                                  0.01 = 10-2
1,000 = 103                                 0.001 = 10-3
10,000 = 104                                0.0001 = 10-4
100,000 = 105                               0.00001 = 10-5
1,000,000=106                               0.000001 = 10-6
10,000,000 =107                             0.0000001 = 10-7
100,000,000 = 108                            0.00000001 = 10-8

A large number can be put into scientific notation by writing it as the first digit of a
number, followed by a decimal point, followed by all of the other digits in the number,
multiplied by a power of 10. The number below is in scientific notation.
8.75 x 108

To convert a number much greater than one to scientific notation, you must do two
things:
1. Move the decimal point in the number to the left so that it is between the left-end
digit in the number and the digit that is next to the left end. If the number does
not have a decimal point to start with, put one on the right-hand end of the
number.
2. Multiply the number by a power of ten. The power should be equal to the number
of digits that you had to pass when you moved the decimal point to the left.

Example 1: Convert the number 8415.7 to scientific notation.

Step 1: Move the decimal point between the 8 and the 4.
8.4157
Step 2: Because you moved the decimal point three digits to the left, multiply by ten to the third
power.
8.4157 x 103

Example 2: Convert the number 532,000 to scientific notation.

Step 1: Since there is no decimal point to start with, place one on the right-hand end of the
number.
532,000.0
Step 2: Move the decimal point between the 5 and the 3.
5.32
Step 3: Because you moved the decimal point five digits to the left, multiply by ten to the fifth
power.
5.32 x 105
Notice that it is not necessary to keep the extra 0s on the right-hand end of the number.
To convert a number much smaller than one to scientific notation, you must do two
things:
1. Move the decimal point in the number to the right so that it is just to the right of
the first digit that is not zero.
2. Multiply the number by a negative power of ten. The power should be equal to
the number of digits that you had to pass when you moved the decimal point to
the right.

Example 3: Convert the number 0.000657 to scientific notation.

Step 1: Move the decimal point between the 6 and the 5.
6.57
Step 2: Because you moved the decimal point four digits to the right, multiply by ten to the negative
fourth power.
6. 57 x 10-4

Example 4: Convert the number 0.0000008 to scientific notation.

Step 1: Move the decimal point to the right of the eight.
8.0
Step 2: Because you moved the decimal point seven digits to the right, multiply by ten to the
negative seventh power.
8.0 x 10-7

Combining Numbers in Scientific Notation
Multiplying numbers in scientific notation

Think about 102 x 103. The meaning of the exponent 2 in 102 tells you that 102 = 10 x 10
and the meaning of the exponent 3 in 103 tells you that 103 = 10 x 10 x 10.
So it must be true that:
102 x 103 = 10 x 10 x 10 x 10 x 10
Therefore, 102 x 103 = 105

To multiply two powers of 10, add the exponents.

Example 5: Calculate (2.7 x 108) (5.0 x 106).
Step 1: Rearrange the numbers to group the powers of ten together.
(2.7 x 5.0) (108 x 106)
Step 2: Multiply the numbers in the two sets of parentheses.
13.5 x 108+6 = 13.5 x 1014
Step 3: Rewrite the number in scientific notation.
1.35 x 1015

Example 6: Multiply 3.567 x 10-4 by 8.5 x 10-3
Step 1: Rearrange the numbers to group the powers of ten together.
(3.567 x 8.5) (10-4 x 10-3)
Step 2: Multiply the numbers in the two sets of parentheses.
30.3195 x 10-4 + (-3) = 30.3195 x 10-7
Step 3: Rewrite the number in scientific notation.
3.03195 x 10-6
Dividing numbers in scientific notation

Think about 105/103. This is the same as 10 x 10 x 10 x 10 x 10/10 x 10 x 10 and this
fraction can be reduced to 10 x 10.
Therefore, 105/103 = 102

To divide one power of 10 by another power of 10, subtract the exponent of the
divisor from the exponent of the dividend.

Example 7: Divide 2.41 x 102 by 7.4 x 1021

Step 1: Find 2.41/7.4 and subtract the exponents.
0.32568 x 10(2-21) = 0.32568 x 10-19

Adding & Subtracting Numbers in Scientific Notation

To add or subtract two numbers in scientific notation:
1. Adjust the powers of 10 in the 2 numbers so that they have the same index. (Tip:
It is easier to adjust the smaller index to equal the larger index).
2. Add or subtract the numbers.
3. Give the answer in scientific notation.

Example 8: Evaluate 2 × 103 + 3.6 × 104, giving your answer in scientific notation.

Example 9: Evaluate 7 × 105 – 5.2 × 104, giving your answer in scientific notation.

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