The laws of indices on one page

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:

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:

• 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

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.