# Compute with scientific notation

This lesson will show you how to compute with scientific notation. You will learn to add, subtract, multiply, and divide with scientific notation.

It is important to understand how to convert numbers to scientific notation before starting this lesson.

## Add and subtract with scientific notation

Example #1:

Add and subtract 2 × 105 + 4 × 103 and -6 × 105 + 2 × 103 + 9 × 104

When adding or subtracting numbers in scientific notation, you can only add or subtract numbers that have the same exponent. Therefore, first step is to put the ones with the same exponent together.

Addition: 2 × 105 + 4 × 103 + -6 × 105 + 2 × 103 + 9 × 104

(2 × 105 + -6 × 105) + (4 × 103 + 2 × 103) + 9 × 104

Add the numbers you see on the left of 105.This is the number before the multiplication sign. Then, put the sum next to 105 and put the multiplication sign between the sum and 105 . Do the same thing for 103

2 + -6 = -4 and 4 + 2 = 6

2 × 105 + 4 × 103 + -6 × 105 + 2 × 103 = -4 × 105 + 6 × 103 + 9 × 104

Subtraction: 2 × 105 + 4 × 103 - ( -6 × 105 + 2 × 103 + 9 × 104 )

2 × 105 + 4 × 103 - -6 × 105 - +2 × 103 - +9 × 104

2 × 105 + 4 × 103 + 6 × 105 - 2 × 103 - 9 × 104

(2 × 105 + 6 × 105) + (4 × 103 - 2 × 103) - 9 × 104

Add the numbers you see on the left of 105 and put the answer next to 105.

Subtract the numbers you see on the left of 103 and put the answer next to 103.

2 + 6= 8 and 4 - 2 = 2

2 × 105 + 4 × 103 - ( -6 × 105 + 2 × 103 + 9 × 104 ) = 8 × 105 + 2 × 103 - 9 × 104

Observation: a × 10n - a × 10n = 0 × 10n = 0

For example, 8 × 107 - 8 × 107 = 0 × 107 = 0

## Multiply and divide with scientific notation

Example #2:

Multiply and divide 64000000 and 0.0008

Multiplication: 64000000 × 0.0008

When multiplying or dividing numbers in scientific notation, it is irrelevant to combine numbers that have the same exponent.

Since 64000000 and 0.0008 are not in scientific notation, put them in scientific notation.

64000000 = 6.4 × 107

0.0008 = 8 × 10-4

64000000 × 0.0008 = 6.4 × 107 × 8 × 10-4 = 6.4 × 8 × 107 × 10-4

Multiply 6.4 and 8. Then, multiply 107 and 10-4

6.4 × 8 = 51.2

107 × 10-4 = 107 + -4 = 103

We get 51.2 × 103

Put 51.2 × 103 in scientific notation

51.2 × 103 = 5.12 × 101 × 103 = 5.12 × 104

Division: 64000000 ÷ 0.0008

64000000 ÷ 0.0008 = (6.4 × 107) ÷ (8 × 10-4)

(6.4 × 107) ÷ (8 × 10-4) =
6.4 × 107 / 8 × 10-4

 6.4 × 107 / 8 × 10-4 =   6.4 / 8 ×   107 / 10-4

 6.4 × 107 / 8 × 10-4 =   6.4 / 8 ×   107 / 10-4

Divide 6.4 and 8. Then, Divide 107 by 10-4

6.4 ÷ 8 = 0.8

107 ÷ 10-4 = 107 - -4 = 1011

We get 0.8 × 1011

Put 0.8 × 1011 in scientific notation

0.8 × 1011 = 8 × 10-1 × 1011 = 8 × 1010

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