Math 1 · Unit 7 · Lesson 1 · part 4 of 6
The zero power, by walking backward
What is 2^0?
The zero power, by walking backward
What is 20? Zero 2s multiplied together sounds like nothing. Do not guess; walk the table backward. Each step to the right multiplies by 2. So each step to the left divides by 2.
| x | 2x | how we got there |
|---|---|---|
| 3 | 8 | 2 × 2 × 2 |
| 2 | 4 | 8 ÷ 2 |
| 1 | 2 | 4 ÷ 2 |
| 0 | 1 | 2 ÷ 2: one more step left |
| −1 | 12 | 1 ÷ 2: and one more |
| −2 | 14 | 12 ÷ 2 |
From 21 = 2, one more step left lands on 1. So 20 = 1. Keep going: 2−1 = 12, and 2−2 = 14. The table keeps going to the left, and the graph does too: the outputs get smaller and smaller, but they are always a fraction of something, never 0.
Multiplying by 1 is doing nothingAny base to the zero power is 1. That is the multiplicative identity from Unit 1, Lesson 1, the do-nothing number, showing up again: at x = 0 the exponential has multiplied by its base zero times, so it is exactly where it started. You will see this secret 1 again in Lesson 2, when it explains the y-intercept.
Reading and patterns only, for now. The rules for multiplying and dividing powers, like what 23 × 24 is, wait for Unit 13. Here you only need to read a power, count the multiplying, and walk a table both ways.