Åk 6–9

6.8 Calculating with the Standard Form
On this page, we will show you the rules of exponents and how to calculate using these rules with numbers in the standard form.

A number written in the standard form is a product that consists of two factors. The first factor is a number between 1 and 10 and the second factor is written as a power of ten. You can read more about this on page 6.3 Standard Form.

Multiplication

Example 1

What  is 4 · 104 multiplied by 2 · 103?

When we solve this problem, we can use the associative property. According to the associative property, it does not matter in which order the factors are written when we multiply them together (See 8.3).

4 · 104 ·  2 · 103 = 4 · 2 · 104 · 103 = 8 · 107

We change the order between the factors so that it will be easier to do the calculations.  Example 2

What is 6 · 10-2 multiplied by 7 · 105?

6 · 10-2  ·  7 · 105 = 6 · 7 · 10-2 · 105 = 42 · 103

We change the order between the factors and make it easier to do the calculation. 42 · 103 does not fulfill the rule that the first factor should be between the number 1 and 10 and we are therefore not finished:

42 · 103 = 4.2 · 10 · 103 = 4.2 · 10 · 103 = 4.2 · 104

4.2 · 104does fulfill the rule that the number is between 1 and 10 and the second factor is a power of 10.  We are finished!

Division

Example 1

What is 6 · 107 divided by 2 · 104?

Even here, we break out the factors between 1 and 10 from the powers of ten.

 6 · 107 2 · 104 = 6 2 · 107 104 = 3 · 107-4 = 3 · 103  Example 2

 2 · 103 8 · 10-7 = 2 8 · 103 10-7 = 0.25 · 103-(-7) = 0.25 · 103+7 = 0.25 · 1010 0.25 · 1010 does not fulfill the rule that the first factor should be between the number 1 and 10 and we are therefore not finished:

0.25 · 1010 = 2.5 · 0.1 · 1010 = 2.5 · 10-1· 1010 =
= 2.5 ·10-1 · 1010 = 2.5 · 109

Addition and subtraction of number is the standard form
When adding and subtracting numbers in the standard form, we can NOT use the rules above.  The easiest way to solve these problems is to rewrite them to normal numbers before doing any calculations.

If the answer is then to be given in standard form, rewrite the results after calculating.

Example

7 · 101 + 2.5 · 103 = 70 + 2,500 = 2,570 = 2.57 · 103 Subtraction

Example

8 · 104 - 6 · 103 = 80,000 - 6,000 = 74,000= 7.4 · 104 