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 · 10^{4} multiplied by 2 · 10^{3}?
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 · 10^{4} · 2 · 10^{3} = 4 · 2 · 10^{4} · 10^{3} = 8 · 10^{7}
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 · 10^{5}?
6 · 10^{2} · 7 · 10^{5} = 6 · 7 · 10^{2} · 10^{5} = 42 · 10^{3}
^{
}We change the order between the factors and make it easier to do the calculation.^{
}
42 · 10^{3 }does not fulfill the rule that the first factor should be between the number 1 and 10 and we are therefore not finished:
42 · 10^{3} = 4.2 · 10 · 10^{3} = 4.2 · 10 · 10^{3} = 4.2 · 10^{4
}4.2 · 10^{4}does 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 · 10^{7} divided by 2 · 10^{4}?
Even here, we break out the factors between 1 and 10 from the powers of ten.
Example 2
0.25 · 10^{10 }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 · 10^{10} = 2.5 · 0.1 · 10^{10} = 2.5 · 10^{1}· 10^{10} =
=
2.5 ·10^{1} · 10^{10} = 2.5 ·
10^{9}^{}
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.
Addition
Example
7 · 10^{1} + 2.5 · 10^{3} = 70 + 2,500 = 2,570 = 2.57 · 10^{3}
Subtraction
Example
8 · 10^{4}  6 · 10^{3} = 80,000  6,000 = 74,000= 7.4 · 10^{4}
