Squares, Cubes, Roots & Indices

An index counts how many times the number is multiplied by itself, so n² is n × n and n³ is n × n × n. Drawing them makes the names literal: squaring gives a square, cubing gives a cube.

Multiplying powers adds the indices. n² × n³ is (n×n) × (n×n×n), which is five ns multiplied, so n⁵. Dividing subtracts them for the same reason. You never have to memorise these laws if you can write the multiplication out.

A negative index means one over, and a zero index means one. n⁻² is 1/n², because dividing keeps subtracting from the index and it must carry on past zero. And n⁰ = 1 follows from nᵃ ÷ nᵃ = 1, which is the same as nᵃ⁻ᵃ.

• Try this: Set n = 3, a = 2, b = 3. What does the product law predict, and does the readout agree?
• Try this: Set b to a negative index. What is nᵇ now, and what happens to the product?
• Try this: Set a = b. What is nᵃ ÷ nᵇ, and what does that prove about n⁰?