Rules of Indices

First indices rule

This is a very popular rule in indices. This is known as the multiplication rule. Powers can be used to indicate how many times a (base) has been multiplied by itself. For example;

…tells us that ‘a’ has been multiplied by itself 5 times. We write this as;

The power ‘5’ is the number of times that a has been multiplied by itself. In the rule we can replace m and n with 4 and 3 respectively;

Since the base is the same in both terms the multiplication is just a continuation. To observe this expand the expression;

The total number of as in the expansion is equal to the sum of powers in the question.

The example below shows where this rule applies.

Example: Workout 2⁵ x 2⁶

Answer

2⁵ × 2⁶ = 2⁵⁺⁶ = 2¹¹

ExplanationWe must use the indices multiplication rule, we simply just add the powers.

It’s is important to leave the answer in indices. 2¹¹ is much easier to remember than 2048.

Leaving the answer in indices form is particularly helpful when working with very large indices. 9⁸ is much easier to remember than 43046721.

We can expand the expression to see exactly what is going on.

2⁵ = 2 × 2 × 2 × 2 × 2

…and…

2⁶ = 2 × 2 × 2 × 2 × 2 × 2

We can continue to multiply the two products together.

2⁵ × 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2¹¹

…therefore…

2⁵ × 2⁶ = 2¹¹

Second indices rule

This rule of indices is known as the power of a power. A number with a power can be raised to a power eg; a⁵ to the power 2.

This expression simply means;

…we know that the first rule tells us that we should add the indices power together for multiplication;

But note also that 5×2 is equal to 10. This suggests that if we have a^m raised to the power n we simply multiply the powers together to get the result a^mxn or simply a^mn, this is proof for the second rule. Below are some examples of how to use this rule.

Example: Work out (4⁵)²:

Answer
The power of two means that we want to have 4⁵ multiplied by itself 2 times. In this case we simply just multiply the powers together.

This example proves the general rule of indices that (aⁿ)^m = a^nm

The answer for the above question is actually a very large number of 1048576 so you can see why it is important that you leave your answers in indices form.

Third indices rule

This rule states that for the division of two powered numbers, the result is equal the base to the power of the difference between the two powers as shown in the following example.

Example: Work out 4⁸ ÷ 4²

Answer: Expand the expression first to observe what is going on.

We can now use cancelling to simplify the expressions.

There has been two cancellations which proves that we only need to find the difference between the two powers as shown below.

Remember to leave your answers in indices, rather than the actual value. This is useful when carrying out much large/complex calculations.

Fourth Indices Rule

This next rule is very obvious (well… to remember). It is not so obvious why any number to the power zero is 1, but there is a number of ways this can be proved. In this article we shall prove it prove it by using the third indices rule.

Fifth Indices Rule

The following indices rule deal with negative and fractional powers.

Look at the pattern below;

By looking at the worked out indices above do you note a pattern?
A negative power on any number creates a reciprocal of that number. You can learn more about reciprocals here.

If you’re to work it out on your calculator you might get one of these variations;

The general rule for negative powers is;

Example: Work out (2/3)^-5;

Answer: We have to work out;

The reciprocal of;

That must mean that;

Next we simply power the denominator and numerator separately as shown below;

Sixth Indices Rule

The indices rule shown above is known as a fractional indices rule. This is the simpler version but is not different from the one shown below.
You must know that anything to the power 1 is itself. So the expression shown below must be true;

The expression above implies that a^½ is the √a. That proves the above rule that;

It also proves that;

…because…

This forms the general indices rule for fractional powers;

Now we shall look at an expansion to this rule. The indices rule explained above is derived from the following rule in fact. We have not powered the final answer with 1 because x to the power of 1 is equal to x.

This is just the harder fractional indices rule.

Example: Workout 27^2/3.

Answer:

Here we could use the brackets rule by first breaking the power into parts.

Thus we could write;


Now we could use the rule for fractional powers.

You can see how this rule and the above are quite similar. We can split up any fractional power into a root followed by a power as shown below;