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 = 10^{7 + -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)



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

6.4 ÷ 8 = 0.8

10⁷ ÷ 10^-4 = 10^{7 - -4} = 10¹¹

We get 0.8 × 10¹¹

Put 0.8 × 10¹¹ in scientific notation

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