When we multiply the same number by itself again and again, writing it out in full gets long fast. Index notation is the shorthand:

We read 2^3 as “two to the power three”. The big number at the bottom is the base — the number being multiplied. The little number up high is the index (also called the exponent or power) — it counts how many times the base appears in the product.

Watch out: a power is not the same as a multiplication. The index tells you how many copies of the base to multiply — not what to multiply the base by.

That is three squared, and it equals 9. It is a common slip to read 3^2 as 3 \times 2 = 6 — but the 2 is the index, not a thing we multiply by. A small index can make a surprisingly big number: 2^{10} = 1024.

Two indices have everyday names: n^2 is “n squared” (because it gives the area of a square with side n), and n^3 is “n cubed”.

Khan Academy introduces exponents here: