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How to Calculate with Scientific Notation

Here, you will take a look at how you do calculations with numbers in scientific notations. It’s a piece of cake. The main idea is to work separately with the power of 10’s and the numbers in front of the powers.

Rule

Multiplication with Scientific Notations

(a ⋅ 10n) ⋅(b ⋅ 10m) = a ⋅ 10n ⋅ b ⋅ 10m = a ⋅ b ⋅ 10n+m

(a ⋅ 10n) ⋅(b ⋅ 10m) = a ⋅ 10n ⋅ b ⋅ 10m = a ⋅ b ⋅ 10n+m

Rule

Division with Scientific Notation

a ⋅ 10n b ⋅ 10m = a b ⋅ 10n 10m = a b ⋅ 10n−m

Example 1

Calculate (4 ⋅ 104) ⋅(2 ⋅ 102)

You use the multiplication rule.

(4 ⋅ 104) ⋅(2 ⋅ 102) = 4 ⋅ 2 ⋅ 104+2 = 8 ⋅ 106

(4 ⋅ 104) ⋅(2 ⋅ 102) = 4 ⋅ 2 ⋅ 104+2 = 8 ⋅ 106

Example 2

Calculate 5 ⋅ 104 ⋅ 3 ⋅ 105

Remember to sort.

5 ⋅ 104 ⋅ 3 ⋅ 105 = 5 ⋅ 3 ⋅ 104+5 = 15 ⋅ 109 = 1.5 ⋅ 10 ⋅ 109 = 1.5 ⋅ 109+1 = 1.5 ⋅ 1010

A value in scientific notation has a number from 1 to 10 multiplied by a power of 10. Since 5 ⋅ 3 =15, you will need to convert 15 to a number between 1 and 10. To do this, write 15 = 1.5 ⋅ 10 (in scientific notations) and repeat once more.

Example 3

Calculate 8 ⋅ 106 4 ⋅ 104

You use the division rule.

8 ⋅ 106 4 ⋅ 104 = 8 4 ⋅106 104 = 2 ⋅ 106−4 = 2 ⋅ 102 = 200

8 ⋅ 106 4 ⋅ 104 = 8 4 ⋅106 104 = 2 ⋅ 106−4 = 2 ⋅ 102 = 200

Example 4

Calculate 3 ⋅ 109 6 ⋅ 105 and write in scientific notation

You factorize into two fractions

3 ⋅ 109 6 ⋅ 105 = 3 6 ⋅109 105 = 0.5 ⋅ 109−5 = 0.5 ⋅ 104 = 5 ⋅ 10−1 ⋅ 104 = 5 ⋅ 103

A value in scientific notation has a number from 1 to 10 multiplied by a power of 10. You must convert 0.5 to a number between 1 and 10. To do this, write 0.5 = 5 ⋅ 10−1 (in scientific notation) and repeat once more.

The math master thinking about scientific notation

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