Chapter 6
Exponents and Roots
Fold a sheet of paper in half, then in half again, and keep going. Each fold doubles the layers: two, four, eight, sixteen. Ten folds means multiplying two by itself ten times, and writing that string of twos out in full is already tedious. The same repeated multiplication turns up in areas, in volumes, and in very large numbers, so it deserves a shorter name. Exponents are that shorthand, and the numbers they reach climb faster than almost anyone expects, which leaves a second question: given the answer, can you find what was multiplied?
What You'll Explore
5 lessons.
- Introduction to Exponents
Two numbers, one full sized and one raised small beside it, can replace a long line of multiplication. What is that small raised number actually telling you to do? You will work out what the compact form stands for, how it is read aloud, and how fast the values it names grow.
- Laws of Exponents
Multiplying two powers of the same base can be done the slow way, by writing out every factor and multiplying. Do that a few times and a pattern starts showing in the exponents themselves. Here you will chase that pattern down and see how far it extends to dividing and to raising a power again.
- Negative and Zero Exponents
Can a factor appear zero times, or a negative number of times? Those sound like nonsense questions, yet the laws from the previous lesson keep producing exactly those forms, and this lesson asks what they must mean if every rule is to keep holding.
- Square Roots
Squaring a number is straightforward. Running that step backward asks the reverse question: what number, multiplied by itself, gives one hundred? Ask the same about ten and no whole number fits, and that gap is what this lesson explores.
- Scientific Notation
The distance to the Sun is about one hundred fifty million kilometers, and the width of a water molecule begins with nine zeros after its decimal point. Miscount those zeros by one and the number is off by a factor of ten. This lesson looks at a way of writing such numbers that makes their size easy to see.