6.3 Adding, Subtracting, multiplying & dividing
Adding & Subtracting
When we add or subtract numbers that are in standard form, we must get rid of the standard form, then we minus the numbers. We would usually then place the answer back into standard form. Let’s have an example:
When we add or subtract numbers that are in standard form, we must get rid of the standard form, then we minus the numbers. We would usually then place the answer back into standard form. Let’s have an example:
A minus question
Multiplying & Dividing
There are two steps for multiplying and dividing numbers that are in standard form. These two steps are:
There are two steps for multiplying and dividing numbers that are in standard form. These two steps are:
1) Multiply or divide the number
1) Apply the laws of indices to the powers of 10. When we are multiplying powers together we add the powers. For example; 104 x 107 = 104+7 = 1011. When we are dividing powers, we subtract the powers. 10 ÷ 105 = 108-5 = 103. (Note: these laws only apply if the base number is the same. In these examples, the base number was always 10. We would not be able to combine the terms if they had different bases e.g. we cannot add powers for this 57 x 64).
Let’s have an example:
We first multiply the numbers together. 5 x 8 = 40.
The next step is to apply the index laws to the powers. 103 x 104 = 103+4 = 107.
We combine both of these terms to get.
However, this answer is not in standard form because 40 is greater than 10 (remember that A must be between 1 and 10: 1 ≤ A < 10). Therefore, we need to re-adjust the answer. We do this by dividing 40 by 10 to get 4 (which is between 1 and 10) and adding 1 to the power of 10. The final answer becomes:
Division examples:
Like the multiplication example, we divide the number first. 2 ÷ 8 = ¼ = 0.25
The next step is to apply the laws of indices. 108 x 10-5 = 108--5 = 108+5 = 1013 (we need to be careful with the signs on the powers. Minusing -5 becomes adding 5).
We combine both of the terms to get.
However, 0.25 is not between 1 and 10, which means that we need to re-adjust the answer. We do this by multiplying 0.25 by 10 and taking one off the power (13 - 1). The answer becomes.