mathsGrade 9-102 min read

The laws of indices on one page

Six rules cover every power you'll meet, and they all come from one idea: an index just counts how many times you multiply. Includes the negative and fractional ones everyone forgets.

An index (a power) is just shorthand for repeated multiplication: $x^4$ means $x\times x\times x\times x$ . Every rule below falls straight out of that idea — you can rebuild any of them by writing the multiplication out longhand if you forget.

The three core laws

When the base is the same:

x^a \times x^b = x^{a+b} \qquad \frac{x^a}{x^b} = x^{a-b} \qquad \left(x^a\right)^b = x^{ab}

Multiplying adds the powers (you're just stacking more lots of $x$ ); dividing subtracts them; a power of a power multiplies. That's it for the everyday cases.

The three everyone forgets

These are where exam marks hide:

x^0 = 1 \qquad x^{-n} = \frac{1}{x^{n}} \qquad x^{\frac{1}{n}} = \sqrt[n]{x}

Anything to the power zero is $1$ . (Divide $x^3$ by $x^3$ : you get $x^0$ , but also $1$ . So they're equal.)
A negative power means "one over". It flips the term to the bottom of a fraction — it does not make the number negative. $2^{-3} = \tfrac{1}{8}$ , not $-8$ .
A fractional power means a root. $x^{1/2} = \sqrt{x}$ , and $x^{2/3} = \sqrt[3]{x^2}$ .

Worked example

\frac{x^5 \times x^{-2}}{x^{4}} = \frac{x^{5 + (-2)}}{x^4} = \frac{x^{3}}{x^4} = x^{3-4} = x^{-1} = \frac{1}{x}

One law at a time, sign by sign.

The base must match before you add powers. $x^3 \times y^2$ does not simplify — different bases stay apart. The index laws only let you combine powers of the same thing.

Last revised 25 February 2025.