Read the shorthand of algebra

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Read the shorthand of algebra

A letter can stand for a number that is unknown or allowed to vary. Writing 3x means 3 multiplied by x, not a two-digit number. The expressions x² and x³ describe repeated multiplication. A fraction bar can represent division by a nonzero denominator, and brackets identify a grouped quantity.

A coefficient multiplies a term. In 3x², the coefficient is 3 and the power applies to x. In (3x)², the brackets make the entire product the base, giving 9x². A coefficient such as 1/2 can be kept as a fraction, preserving an exact value.

Write expressions so another reader can recover the intended operations. The statement ab means a × b; a/b requires b ≠ 0. The expressions a+b and ab are different even when an unusual choice of numbers happens to make their values equal.

Worked example: Write x+x+x and a×a×b in compact notation.

Three identical x terms combine to 3x.

Two factors of a give a²; retain the separate b factor.

Answer: 3x and a²b.

Check your understanding: Are 3x² and (3x)² generally equal?

Answer: No: the second is 9x².

Why: In the second expression the square applies to the coefficient as well as x.