index notation

The index, or power of a number indicates how many times the number is to be multiplied by itself. For example, 4² (four squared) = 4 × 4 = 16; 2³ (two cubed) = 2 × 2 × 2 = 8; and 5⁴ (five to the power four) = 5 × 5 × 5 × 5 = 625. In these examples, 4², 2³, and 5⁴ are all written in index notation.

There are three important rules for the use of indices:

(i) a^m × aⁿ = a^m+n

(ii) a^m ÷ aⁿ = a^m−n

(iii) (a^m)ⁿ = a^mn

In all these three rules, m and n are assumed to be positive whole numbers and in rule (ii), a is not zero and m > n. For example:

3³ × 3⁴ = 3³⁺⁴ = 3⁷; 5⁴ ÷ 5² = 5⁴⁻² = 5²; (2³)⁴ = 2^3×4 = 2¹²

When using fractional indices, x^1/n = ⁿ√x, so x^1/2 = ²√x and x^1/4 = ⁴√x; also x^m/n = (ⁿ√x)^m or ⁿ√(x^m). The same rules apply when using fractional indices as for whole number indices.