Look at 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2. Each step down divides by 2.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
So what is 2⁰
Continuing the pattern, 2 ÷ 2 = 1. Every non-zero number to the power nought is 1.
Keep going below nought
The next step is 1 ÷ 2 = ½, so 2⁻¹ = ½. And 2⁻² = ¼.
The rule this gives
A negative exponent means one over the positive power. It never makes the answer negative.
Continue the pattern downwards
Consider 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2. Each step down halves. Continuing gives 2⁰ = 1 and 2⁻¹ = 1/2. The values are forced by the pattern, not chosen for convenience.
Why anything to the power zero is 1
By the division law, x³ ÷ x³ = x⁰. But anything divided by itself is 1. So x⁰ must be 1. The one exception is 0⁰, which is left undefined because the two arguments give different answers.
What a negative exponent means
x⁻ⁿ = 1/xⁿ. A negative exponent means a reciprocal, not a negative number: 2⁻³ = 1/8, which is positive. Reading a negative exponent as producing a negative value is a persistent and easily avoided error.
The laws still apply
Every exponent law continues to work with zero and negative exponents — that consistency is exactly why those values were defined as they are. A definition that broke the laws would be worse than useless.
Step 2: Try It Yourself
Halve a quantity repeatedly, which is what each downward step does.
Before: 8
After: 4
The factor is less than 1, so the bar got shorter. Multiplying does not always make things bigger.
Step 3: Watch an Example
One step at a time.
Watch Ana Reach a Negative Power
Ana continues the pattern for base 3.
- Step 1
She writes 3² = 9 and 3¹ = 3.
Step 4: Your Turn
Practice makes it stick.
The Zero Power
Problem 1 of 2
What is 7⁰?
The Negative Power
Problem 2 of 2
What is 2⁻²? Give it as a decimal.
Down the Pattern
1 of 4
What is 5⁰?
2 of 4
What is 10⁻¹? Give it as a decimal.
3 of 4
What is 4⁻¹? Give it as a decimal.
4 of 4
Is 2⁻³ a negative number? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
What is 10⁻²? Give it as a decimal.
What You Learned
- Each step down the exponent divides by the base.
- That forces any non-zero number to the power nought to be 1.
- A negative exponent gives a reciprocal, never a negative answer.