Make Your Own Rules

We are going to need two more important properties of indices. But now that you have seen how to work things through you should be able to work these rules out for yourself. Don't just guess. Use the definitions and the properties of multiplication, division and fractions just like we have just done.

In the following table click on the questions and answers to see the hints and answers. Click on to see a fully worked answer.

	Question	Answer
1.
2.
3.

These problems all involve expansion of bracketed expressions raised to powers. The bracketed expressions involve multiplication and division only. Here is a formal statement of the rules.

Expansion Rules for Brackets and Indices.

If and are any real numbers and is a positive integer then

2. , provided

Not the Expansion Rule

The expansion rule works for multiplication and division only. This is because indices represent repeated multiplication. In particular the rule does NOT work for addition and subtraction. This is one of the most common student mistakes. Here is what it looks like.

It doesn't work because it mixes up addition and multiplication and you just can't change the order of these and get the same answer.

An Interesting Case

The three laws of indices are about powers with common bases. These expansion rules are about powers with common indices. An interesting situation arises when both the bases and the indices are equal. In this case you can use either of the possible rules, but not both of them at the same time. Try and work it out yourself.

	Question	Answer
Common base
Common index
NOT both