Chapter 2: Power Play

The Earth weighs roughly $5{,}972{,}000{,}000{,}000{,}000{,}000{,}000{,}000$ kilograms. The diameter of a hydrogen atom is about $0.000{,}000{,}000{,}1$ metre. Writing such numbers out by hand is tedious, error-prone, and easy to misread. Mathematicians solved this nuisance long ago with a small but mighty piece of notation: the exponent.

An exponent compresses a long product into a short symbol. Instead of $10 \times 10 \times 10 \times 10 \times 10 \times 10$ we write $10^6$. Instead of $\frac{1}{10 \times 10 \times 10}$ we write $10^{-3}$. With these abbreviations the Earth's mass becomes $5.972 \times 10^{24}$ kg and the hydrogen atom is $1 \times 10^{-10}$ m. Both fit comfortably on a single line.

But exponents are not just compact notation. They follow a beautiful set of rules , the laws of exponents , that let us combine, simplify, and compare any two powers without ever writing them out. These rules will appear again and again in algebra, science, and finance.

By the end of this chapter you will: read and write numbers in scientific notation (also called standard form); apply the laws $a^m \cdot a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, $a^0 = 1$ and $a^{-n} = 1/a^n$ to simplify expressions; and use these tools to compare astronomically large and microscopically small numbers.

What's inside

1. Meet exponents , base, exponent, and what they really mean.
2. Laws of exponents , the six rules you will use forever.
3. Zero and negative exponents , what $a^0$ and $a^{-n}$ mean.
4. Scientific notation (standard form) , writing very large and very small numbers.
5. Comparing and using powers in the real world , populations, distances, and tiny things.

Key results

Standard form. A number $N$ is in standard (scientific) form if $N = k \times 10^n$ where $1 \le k < 10$ and $n$ is a whole number (positive, zero, or negative). Examples: $3{,}840{,}000 = 3.84 \times 10^6$; $0.00072 = 7.2 \times 10^{-4}$.

How to read this chapter

Keep a calculator nearby to verify simplifications, but try every problem by hand first. Most "tricky" exponent questions become easy once you remember that an exponent is just repeated multiplication (or repeated division for negatives).