Indices, a shortcut for repeated multiplication

Squares and cubes are nothing new, let’s explore what happens when the values grow and shrink

Your learner will be very familiar with formulas like:

$A=lb$

$V=lbh$

$A=πr^2$

In these examples, the units that you use can end up being $cm^2$ or $m^3$.

The power used for the units tells what the dimensions are for the answer. The second power involves length and breadth, and the third power makes use of length, breadth, and height.

Indices is the fancy word for powers.

Using them isn’t much different than thinking about the multiplication you use to calculate area or volume.

But there are some rules to learn and knowing them will take a load off your learner’s shoulders.

The index rules

In total, there are six index rules.

The suggested order is:

1. $x^a\times x^b=x^{a+b}$
2. $x^a\div x^b=x^{a-b}$
3. $(x^a)^a =x^{a\times a}$
4. $x^{-1}=\frac{1}{x}$
5. $x^\frac{1}{2}=\sqrt{x}$
6. $x^0=1$ and $x^1=x$

Let’s start with rule 1

$3^2\times 3^3=3^{2+3}=3^5$

But surely that looks too easy…

Let’s test it by expanding the terms:

$3^2=3 \times3=9$

$3^3=3 \times3\times3=27$

$9\times27=243$

$3^5=3\times3\times3\times3\times3=243$

Maybe that just works for 3s.

$x^4\times x^5=x^{4+5}=x^9$

$x^4=x\times x\times x\times x$

$x^5=x\times x\times x\times x\times x$

When you multiply 4 $x$s by 5 of them, you will get 9 $x$s in an expression or $x^9$.

This works every time.

Rule 2 is illustrated below:

$2^7\div2^4=2^{7-4}=2^3$

Again, this looks too easy, there must be more to it.

To illustrate the rule, remember that division is like simplifying a fraction.

$2^7\div2^4=\frac{2^7}{2^4}$

Next, expand the numerator and denominator:

$\frac{2^7}{2^4} = \frac{{2} \times \sout{2} \times \sout{2} \times \sout{2} \times 2 \times 2 \times 2}{\sout{2} \times \sout{2} \times \sout{2} \times \sout{2}}$

You will see the denominator has 2, five times, which can be divide the five 2s in the numerator:

$2\times2\times2=2^3$

And as you have recently read, dividing with terms isn’t different from dividing with numbers.

$y^8\div y^5=y^{8-5}=y^3$

$y^8\div y^5=\frac{y\times y\times y\times y\times y\times y\times y\times y}{y\times y\times y\times y\times y}=y\times y\times y=y^3$

For Emerging Learners

Get used to expanding the powers until learners start to connect the rule with the process.

Once this procedure is understood it will be difficult to forget, and it will have a significantly positive impact on all future algebraic learning.

There are four rules still to go, so get these two secure through practice.

For learners needing extension

Challenge your learner with, “What do you think will happen if I multiply and divide powers in the same question?”

Explore more complex expressions where you might be multiplying terms followed by dividing terms with different powers.

Or, have a numerator where there is multiplication of terms and include a denominator.

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

• Simplifying fractions
• Expanding indices through repeated multiplication

Multiplying and dividing with indices will soon become commonplace, with Nat 5 maths.

Starting with a slow and steady pace, introducing a couple of new rules a week, is a sensible way forward.

As the weeks progress, take the occasional opportunity to revisit learned content to check it is still remembered and, more importantly, understood.

The next couple weeks will have indices as a focus.

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