Use of symbols

• Letters stand for numbers: an unknown in an equation, or a variable in an expression or formula.
• Algebra follows the rules of arithmetic: a + b = b + a, ab = ba, a(b + c) = ab + ac.
• a³ = a × a × a; a⁰ = 1; a⁻ⁿ = 1/aⁿ.
• Index laws: x^m × xⁿ = x^m+n, x^m ÷ xⁿ = x^m−n, (x^m)ⁿ = x^mn.
• Higher: fractional indices, a^1/n = ⁿ√a and a^m/n = (ⁿ√a)^m.

Expressions, equations, formulae and identities

A letter in algebra always represents a number. What it means depends on where it appears:

Because letters are numbers, algebra follows the generalised rules of arithmetic: order does not matter in addition or multiplication (a + b = b + a, ab = ba), brackets multiply out (a(b + c) = ab + ac), and subtraction and division are not commutative (a − b ≠ b − a in general).

Positive, zero and negative powers

An index (power) counts how many times a number is multiplied by itself: a³ = a × a × a. The pattern 2³ = 8, 2² = 4, 2¹ = 2 halves each time, which forces 2⁰ = 1, 2⁻¹ = ½, 2⁻² = ¼.

Multiply, divide, raise to a power

The laws work only when the base is the same. Deal with the numbers (coefficients) and each letter separately.

Roots written as powers