Powers & Roots

In this video, we are going to go over Powers and Roots. Powers are often referred to as Indices like when we deal with BIDMAS – however, there will be another video on BIDMAS separately.

The key to getting super comfortable with any and all Powers and Roots comes down to remembering some very simple rules.

Let’s start with the basics and build from there:

Rule 1 – Dividing Powers

When we are dividing powers like \(3^4 \div 3^2\), we just subtract the powers. So our answer would be \(3^2.\) Similar to Rule 1, the big number needs to be the same on both sides of the division sign.

Rule 2 – Multiplying Powers

When we are multiplying powers like \(2^2 \times 2^3\), we just add the powers together. So our answer would be \(2^5\). The important thing to note is that the big number must be the same i.e. 2 multiplying 2. If we were to have \(2^2 \times 3^3\), it wouldn’t work.

Rule 3 – Power to Itself

This is a super simple one but anything to the power of 1 is just itself. e.g. \(23^1 = 23\).

Rule 4 – Raising 1 Power To Another

When we apply a number that already has a power, to another power i.e. \((2^3)^4\), you just multiply the powers together. In this example, it would be \(2^{12}\) as $2^{3 \times 4 = 12}$.

Rule 5 – Powers and Fractions

This sounds a little daunting but you just apply the power to the numerator and the denominator. An example would be \((\frac{1}{2})^3\), you could just rewrite this as \(\frac{1^3}{2^3}\).

Rule 6 – Negative Powers

A little more advanced that dividing and multiplying powers but the concept is really simple. Whenever we have a negative power, we just flip the big number and make the power positive. Sounds a little complicated but is super straight forward on paper.

Say we have \(3^{-4}\), at first glance, it’s a bit disgusting. But if you remember to flip and make it positive, it will look something like this:

First – flip the big number —> \(\frac{1}{3}\)

Second – make the power a positive —> \(\frac{1}{3}^4\)… that’s it.

Rule 7 – Powers that are fractions

This could be something like \(4^\frac{1}{2}\). The way I like to remember how this works is to imagine the fraction is like a tree – it seems a bit farfetched but you’ll see why in a second.

If you imagine the line between the numerator and the denominator is like the ground. Anything above the ground would be a strong, powerful trunk and anything below the ground is the roots.

So as you may have guessed, the numerator would dictate the power to apply to the number and the denominator would dictate the root to apply to the number.

In this example, \(4^\frac{1}{2}\) represents \(\sqrt{4}\) as a square root is the same as the root of 2. Therefore our answer is 2.

Rule 8 – Powers that are fractions pt.2

Now, say you have something like \(4^\frac{3}{2}\) – we know from rule 7 that we need to apply 4 to the power of 3 and also apply it to the root of 2. But which do we do first?

Simple, work from the ground up. You start with the root – being 2 so we do \(\sqrt{4}\) which is 2. Then we apply it to the power of 3 (or cubing it).

\(2^3 = 8\), therefore \(4^\frac{3}{2} = 8\).

Hopefully this all made sense. If you’ve got any questions, head over to the forum where you can read up on any questions someone else has asked – or ask your own question!