Why negative and fractional powers work the way they do

Most of the students I help have a pretty good grasp of the three straightforward power laws:

$(x^a)^b = x^{ab}$
$x^a \times x^b = x^{a+b}$
$x^a \div x^b = x^{a-b}$

So far, so dandy – and usually good enough if you’re hoping for a B at GCSE. The trouble comes when they start throwing strange things in: what’s $3^{-2}$? Or $81^{\frac14}$? Or, for the love of all that’s holy, $16^{-\frac32}$? How on earth do you multiply something by itself negative two times? Or a quarter of a time?

Non-positive powers

Non-positive numbers are probably the easier of the two to get to grips with, and I have two ways to explain them. The first involves making a list:

$10^3 = 1,000$
$10^2 = 100$
$10^1 = 10$ … you see how it’s dividing by 10 each time? That pattern continues:
$10^0 = 1$
$10^{-1} = \frac{1}{10}$
$10^{-2} = \frac{1}{100}$
… and so on. In general, $x^{-k} = \frac{1}{x^k}$ – the negative power just ‘flips’ whatever you’re working with and turns it into a fraction.

That means $3^{-2} = \frac{1}{3^2} = \frac19$; similarly, $2^{-6} = \frac{1}{2^6} = \frac{1}{64}$.

The second argument is that $3^{-2}$ must be the same as $3^{0-2} = 3^0 \div 3^2 = \frac{1}{9}$. Easy!

Non-integer powers

Fractional powers are a bit harder to get your head around, but they do make sense – fractions, remember are really division sums. Division sums are the opposite of multiplications.

Remember that $x^{ab} = (x^{a})^b$? Well, it stands to reason – since roots are the opposites of powers – that $x^{\frac ab}$ is the same as $\sqrt[b]{x^a}$.

So, to work out $81^\frac14$, you need to work out the fourth root of 81. 81 is $9^2$, or $3^4$, so $81^\frac14 = 3$.

In the same vein, $8^\frac23 = \sqrt[3]{8}^2 = 2^2 = 4$.

Combining the two

And how about when they’re combined? Well, you break it down into small steps. If you’ve got $16^{-\frac32}$, you deal with the ugliest thing first: the bottom of the fraction. That means ‘square root’, so you’re left with $4^{-3}$. Already looking better! $4^3 = 64$, so you’ve got $64^{-1}$; the power of negative one is just the reciprocal – so your answer is $\frac{1}{64}$.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

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