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Indices Help

This is the second post in a series of posts which I’m writing to appear on a website for our A level students. You can see the first one, on fractions, here. After sharing the first one I got some great feedback and have some areas to improve it, if you see anything missing from this, or feel I need to clarify / remove anything, please let me know.

Indices, or index numbers. That’s those little numbers that appear raised just after a number or a letter. You know, the “2” that means squared, the “3” that means cubed, etc. They are quite often referred to as “powers”. They are sometimes written like this 3^2 means 3 squared. The power is called the index number (in this case 2) and the number being raised is the base number (in this case 3). Indices are pretty important in mathematics, and you will need to be very well versed in them to be successful on the A level maths course.

What are they all about?

In the first instance, they are the numbers of times a number is multiplied by itself. 2^2 means 2 x 2, 2^3 means 2 x 2 x 2, etc. This means that 2^1 is just 2. Which is something some people forget, don’t be one of those people. Repeat after me: “anything to the power 1 is itself, anything to the power 1 is itself….” 

But what do we do with them?

We need to be able to handle indices. We will save ourselves a lot of time later if we can simplify expressions using them, and it is essential for calculus. So here are “The Rules”, but remember, they only work with the same base number (or letter)….

Rule 1

The first rule of indices is, you don’t talk about indices…. Sorry, I couldn’t resist. Is Fight Club even a film that people still watch?

Rule 1 is: “When multiplying, add the powers“. (I say rule 1, that’s what I call it, not its official name…)

This rule is fairly intuitive. If you have 2^2 x 2^3 then you have (2 x 2) x (2 x 2 x 2) which is the same as 2 x 2 x 2 x 2 x 2, which is by definition 2^5. Try some yourself and see.

Rule 2

Again, this is fairly intuitive. The rule is: “When dividing with indices, subtract powers.

So 3^3/3^2 = 3

This is because we are “cancelling” common factors. Taking this as a fraction we’d have a numerator of 3 x 3 x 3 and a denominator or 3 x 3, so we divide top and bottom to give 3/1 which is just 3.

Rule 3

This rule involves raising a power to a power. Consider the problem (x^3)^2,  this means x^3 multiplied by itself, which gives x^6 (using rule 1). The “shortcut” here is to notice that because of the way multiplication works, “when raising a power to a power you multiply“.

A real common mistake on this type of problem occurs when you get something of the form (2y^4)^3. Often people will evaluate that as 2y^12, but that’s wrong. Don’t be one of those people.

(2y^4)^3 = 2y^4 x 2y^4 x 2y^4

So you get 8y^12.


Rule 4

Negative powers are the reciprocal of positive powers

This follows from rule 2, 3^4/3^6 gives 3^(-2) but if we cancel common factors it gives 1/3^2, hence they are the same. This one is extremely important when we get to calculus.

Rule 5

Fractional powers are roots

Think of it like this, 9^(1/2) x 9^(1/2) = 9^1 (using rule 1). Well 9^1 = 9 so 9^(1/2) multiplied by itself is 9. And we know 3 multiplied by itself is 9. This follows for all square roots. By the same logic we can see that a power of 1/3 is a cube root, a power of 1/n is an nth root, etc. This is quite important for calculus.

Another thing you need to be able to do with fractional roots is evaluate them, and I don’t mean just unit fractions. You need to be able to evaluate stuff like 32^(3/5).

This isn’t as hard as it looks, you just need to tackle it in stages. Split the fraction up using rule 3:

32^(3/5) = (32^(1/5))^3 always do the root first, it makes the number easier to deal with.

In this case 32^(1/5) = 2 and 2^3 = 8 so 32^(3/5) = 8. Questions like this do come up in a level papers, and they come up in the non calculator c1 paper, so it’s handy to know the first 10 powers of 2, 3 and 5. If you are struggling to remember them in an exam, you can work them out and write them down.

The final note on fractional indices is that when they are involved in problems using the other rules you deal with then in the same was as any fraction problem. See this page for help in fractions.

Rule 6Anything to the power 1 is itself” – as mentioned before.

Rule 71 to any power is 1

This is straightforward. 1×1=1 no matter how many times you repeat it.

Rule 8 – “Anything, except 0, to the power zero is 1

This is one people sometimes struggle to get their heads around. An nice way to think of it is this: 2^2/2^2 = 2^0 but 4/4=1. You can try with any base and any power, this always works. The only exception comes when the base is 0. 0^0 is undefined (or indeterminate), in the same way that dividing by zero is.

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