Exponents & logarithms

Repeated multiplication

Multiplication is repeated adding: $4 \times 3 = 4 + 4 + 4$. A power (or exponent) is repeated multiplying: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$. The small raised number says how many copies to multiply.

Three rules fall out of the picture:

• $2^a \times 2^b = 2^{a+b}$ — multiplying powers adds the counts (3 twos times 2 twos is 5 twos).
• $2^0 = 1$ — multiplying zero copies leaves you where you started: 1.
• $2^{-1} = \tfrac{1}{2}$ — a negative exponent means divide instead of multiply.