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:

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

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 ${81}^{\frac{1}{4}}$? Or, for the love of all that’s holy, ${16}^{-\frac{3}{2}}$? 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 $3^{-2}=\frac{1}{3^2}=\frac{1}{9}$; similarly, $2^{-6}=\frac{1}{2^6}=\frac{1}{64}$.

The second argument is that $3^{-2}$ must be the same as $3^{0-2}=3^0÷3^2=\frac{1}{9}$. 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 $x^{\frac{a}{b}}$ is the same as $\sqrt[b]{x^a}$.

So, to work out ${81}^{\frac{1}{4}}$, you need to work out the fourth root of 81. 81 is $9^2$, or $3^4$, so ${81}^{\frac{1}{4}}=3$.

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

Combining the two

And how about when they’re combined? Well, you break it down into small steps. If you’ve got ${16}^{-\frac{3}{2}}$, 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 $\frac{1}{64}$.