Indices and Standard Form

The laws of indices

An index (or power) tells you how many times a base is multiplied by itself; in 2⁵ the base is 2 and the index is 5. When you combine powers of the same base, the following rules save a great deal of work:

When you multiply, add the powers; when you divide, subtract them; and a power raised to a power multiplies. A negative index means a reciprocal and a fractional index means a root. For example 2³ × 2⁴ = 2⁷ = 128, 3⁻² = 1/3² = 1/9, and 16^½ = √16 = 4. These laws only apply when the base is the same, so you cannot combine 2³ and 3² by adding indices; you must work each one out separately first.

Standard form

Standard form (scientific notation) is a compact way to write very large or very small numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. A large number gives a positive power and a small number gives a negative power, so 45000 = 4.5 × 10⁴ and 0.0032 = 3.2 × 10⁻³. To multiply or divide numbers in standard form, deal with the numbers and the powers of ten separately. Standard form is used throughout science, for instance the speed of light, about 3 × 10⁸ m/s, or the diameter of an atom, about 1 × 10⁻¹⁰ m.