When familiar symbols do unfamiliar things

Two index rules that do not do what you might expect

Rules three and four use symbols that your learner will already be familiar with.

Rule three just seems to have an extra index on the outside of a bracket.

Rule four looks exactly like a fraction.

In the words of Brave Sir Robin, “That’s easy!”

But, alas, the Mighty Tim reveals there is more to rules three and four than meets the eye.

Rule 3

The third index rule states:

$(x^a)^b=x^{ab}$

If it looks like taking a power to a power means you multiply the powers, you would be correct.

But why is this?

Let’s break down what it means to take something to a power:

$2^2=2×2$

$3^4=3×3×3×3$

In both instances the base, 2 in the first case and 3 in the second, is multiplied by itself as often as the index tells us.

$x^2=x×x$

This is no different from above, so…

$(x^6)^3=x^6×x^6×x^6=x^{18}$

This process takes the base, which is $x^6$ and multiplies it by itself three times.

Or you can think of it as taking the index and multiplying it by 3.

You’ll notice that to make sense of the third rule we used the first rule: when you multiply numbers that have an index with the same base, we add the powers.

Your learner needs to make use of what they have learned to continue to progress with their maths this year.

Rules one and three might only be a few days apart, but they will come up regularly.

The reality is that rule three is just an expansion of rule one: it isn’t something brand new.

Rule 4

Now is where the fun really starts.

What would you think of when you see $x^{-2}$?

A common first reaction is $-x^2$.

But the best way to challenge a misconception is an established pattern:

$x^3=x×x×x$

$x^2=x×x$

$x^1=x$

$x^0=1$ (we’ll dig into this more, later)

Now for the new part, but is it new? I’ll leave that with you.

$x^{-1}=\frac{1}{x}$

$x^{-2}=\frac{1}{x}×\frac{1}{x}=\frac{1}{x^2}$

The goal is to notice the change: as you decrease the index, you are dividing by the base.

$x^2=x^3÷x=x^2$

$x^1=x^2÷x=x$

$x^0=x^1÷x=1$

$x^{-1}=x^0÷x=\frac{1}{x}$

$x^{-2}=\frac{1}{x}÷x=\frac{1}{x^2}$

Getting used to this pattern will help confidence in your learner grow until the process becomes natural.

For Emerging Learners

Practise by looking at numbers with powers, instead of variables.

$2^2=4$

$2^1=2^2÷2=2$

$2^0=2^1÷2=1$

$2^{-1}=2^0÷2=\frac{1}{2}$

$2^{-2}=2^{-1}÷2=\frac{1}{2}÷2=\frac{1}{4}$

Talk through with your learner what changes each time the index decreases by 1.

Try this again for powers of 3, 4, or 5.

Once you get the hang of this pattern, start generalising with variables.

For learners needing extension

Extend your learner, like you did with the 1^st and 2^nd rules by combining things:

What would happen with the following expression:

$\frac{(x^4×x^2 )^3}{x^6×x^{-2} }$

That looks like a mouthful, but when your learner approaches it one step at a time, they will make light work of it in no time.

At this point in S3, it’s helpful to focus on:

• Index rules one to four
• Breaking brackets and simplifying expressions
• Investigating the impact of indices on brackets

Once again, your learner will see many familiar themes inside a complex wrapper.

The best approach is to encourage them to look for what they know and see how the new concept might have changed an outcome, just a little.

 There will be one more issue discussing indices coming up next week.

After this, we move on to surds, which follows on perfectly after index rule six.

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