The Negative Exponent Flip
A negative power doesn't mean a negative number. It means 'move to the bottom'.
Equation
y = x^(-1)
Graph
Table
| x | y |
|---|---|
| 1 | 1 |
| 2 | 0.5 |
| 3 | 0.33 |
| 4 | 0.25 |
| 5 | 0.2 |
What this lesson covers
What you do
You shape the function y = x^(-n) and watch the graph answer.
Challenges to clear
- Slide n to 2: x⁻² means 1/x², 'move to the bottom'. The x = 2 row must show 0.25.
- Further right it sinks: x = 4 shows 1/16 = 0.0625 — small, but never negative.
- Move three factors to the bottom: slide n to 3, so the curve is a ÷ x³.
- Now the top: x = 2 must read 2.5. Since 2³ = 8, a is 20 — a negative exponent shrinks, but never turns the value negative.
Check yourself
Why does it matter that the BASE is the SAME in the negative exponent law?
Evaluate using the law: 2^(-3) = 1 / 2^3 = 1 / ?
- Because the law x^(-n) = 1/x^n only applies when the base x is identical on both sides. — correct
- Because negative exponents only work with the number 2.
- Because if the bases are different, you must add the exponents instead.
- Because the base determines if the result is positive or negative.
- 6
- -8
- 8 — correct
- 1/6
Think about it
- Will x⁻² ever be negative? What is 1/2²?