Power Play
This chapter explores one of mathematics’ great labour-savers: collapsing repeated multiplication into a single compact power. We open with a thin strip of foil that, doubled in thickness just 46 times, would shoot past the Moon, and let that jolt of surprise carry us into exponential notation, the idea of a base lifted to an exponent. From there we build the laws of exponents (the rules for multiplying, dividing, and raising a power to a power), confront zero and negative exponents, learn to wrangle gigantic and tiny numbers using scientific notation and powers of ten, and contrast linear with exponential growth. Along the way we tour a lantern-maker’s workshop, watch water hyacinths double across a tank, and follow number-names from an ancient Buddhist manuscript right up to the googol.
Learning Outcomes
By the end of this chapter, you will be able to:
- write repeated multiplication in exponential form n^a, naming the base and the exponent, and break numbers into products of prime powers;
- apply the laws of exponents, n^a \times n^b = n^{a+b}, \ n^a \div n^b = n^{a-b}, \ (n^a)^b = n^{ab}, \ m^a \times n^a = (mn)^a, with ease;
- make sense of the zero exponent (n^0 = 1) and negative exponents (n^{-a} = \tfrac{1}{n^a});
- record large and small numbers in scientific notation x \times 10^y and read them with confidence;
- distinguish linear (additive) from exponential (multiplicative) growth, and lean on powers of ten to sense the scale of enormous quantities.