1.6 A) Indices Introduction
Powers are also known as an index (plural indices). They are the numbers that appear next to and above a number or letter. They appear in what is known as the superscript.
Here is a number in index form.
The normal sized number is the base; 4 is the base in the example above. The number in the superscript is known as the power or index; 3 is the power or index in the example above.
The power tells us how many times the base is multiplied by itself. In the example above, the power was 3, which means that 4 is multiplied by itself 3 times. We can rewrite the number above as:
Therefore, 43 is 64.
We use indices because they are a shorter way of writing numbers. For example, we could write 27 in two ways. The first is to keep it as an index. The other way is to write it out in full, which would be 2 x 2 x 2 x 2 x 2 x 2 x 2. Writing this number as 27 is much shorter than 2 x 2 x 2 x 2 x 2 x 2 x 2 and therefore, considerably easier to deal with.
If we have a letter rather than a number, the process is exactly the same. We multiply the unknown by itself the number of times that the power tells us. For example, if we had A5, we could rewrite that as A x A x A x A x A.
Writing a Number in Index Form
If you have a number that has been multiplied by itself a certain number of times, we can rewrite it in index form. For example, rewrite what is below in index form:
The base for our index will be B because B is being multiplied by itself. The power is going to be 3. This is because B is being multiplied by itself 3 times. We are able to write B x B x B as B3.
When we are writing numbers in index form, we need to ensure that the bases of the numbers are the same. In the above example, we were multiplying B by itself 3 times and this meant that we could write it as B3. If the unknowns that were multiplied were not the same, we would be unable to combine them and write them in index form.
For example, write what is below in index form wherever possible.
There are two different unknowns in the multiplication above (C and G), which means that we are unable to combine all of the terms together because we have different bases. However, we are able to combine all of the C’s and all of the G’s seperately. There are 3 C’s multiplied by each other and this means that we are able to write these as C3. We are then able to multiply this by the G. This means that the answer in index form wherever possible can be written as C3 x G (or C3G).
There are a few rules when dealing with indices.
Anything to the Powers of 1 is Itself
The first rule is that anything to the power of 1 is itself. This is because we are multiplying the base once. For example:
Anything to the Power of 0 is 1
The second rule is that anything to the power of 0 is 1. For example:
1 to the Power of Anything is 1
The final rule is that 1 to the power of any number is 1. This rule comes about because 1 multiplied by any number of 1’s is always going to be 1. For example:
