Powers and Indices

Indices are a short way to write repeated multiplication. A power has two parts: the base, the number being multiplied, and the index, the small raised number that says how many times to use it. So 3⁴ means 3 × 3 × 3 × 3, read “3 to the power of 4”. The base and the index cannot be swapped: 4³ is a different number.

Nearly. A power is the whole expression, such as 3⁴, made of a base and an index. The index is just the small raised number, so it is one part of a power rather than another word for it. In American English the index is called the exponent; indices, exponents and powers all describe the same idea.

A square number is a whole number multiplied by itself, a number to the power of 2: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 are the square numbers up to 100. So 7 and 20 are not square numbers, because no whole number squared gives them. Cube numbers use the power of 3: 1, 8, 27, 64 and so on.

A cube number is a whole number multiplied by itself twice, a number to the power of 3: 2³ = 2 × 2 × 2 = 8, so 1, 8, 27, 64 and 125 are the first five cube numbers. Square numbers use the power of 2 instead: 1, 4, 9, 16, 25.

Any number except 0 raised to the power of 0 equals 1, so 10⁰ = 1 and 5⁰ = 1. It is worth remembering alongside the other special rule: any number to the power of 1 equals itself, so 9¹ = 9. The one exception is 0⁰, which is undefined.

Follow BIDMAS: work out brackets first, then indices, then multiplication. In (2 × 5)², the brackets give 10, and 10² = 100. In 2 × 5², there are no brackets, so square first: 5² = 25, then 2 × 25 = 50. The two answers are completely different, so the order matters.

Build squares with counters: 2 rows of 2 make 4, 3 rows of 3 make 9, 4 rows of 4 make 16. Each is a square shape, which is why they are called square numbers, and the maths is 2 × 2, 3 × 3, 4 × 4. Then show the short way to write it: 4².