Zero and Negative Exponents
a⁰ = 1 and a⁻ⁿ is the reciprocal of aⁿ.
About this topic
Any non-zero number raised to the power zero equals 1, which follows from the quotient rule: xᵃ / xᵃ = x⁰ = 1. A negative exponent means a reciprocal, so x⁻ⁿ = 1/xⁿ. For example 2⁻³ = 1/8.
A negative exponent does not make the answer negative; it moves the base to the other side of the fraction bar. So 3x⁻² = 3/x², and 1/y⁻⁴ = y⁴. Think of the pattern 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2: each step down divides by 2, so the sequence continues naturally into fractions.
Negative exponents are used constantly in science. Scientific notation writes 0.00045 as 4.5 × 10⁻⁴, and units such as meters per second squared are written m · s⁻². Chemists and physicists use them to handle tiny sizes like atoms and bacteria.
Students sometimes believe that 0⁰ or x⁰ with x = 0 is 1, but zero to the zero power is left undefined in this course. Also be careful: in −3⁰ only the 3 has the exponent, so the answer is −1, while (−3)⁰ equals 1. A negative exponent affects only the symbol directly before it.
Key ideas
- Any nonzero base to the power 0 is 1
- Negative exponent means flip to the denominator
- 2⁻³ = 1/8
Quick check
Use cases: where Zero and Negative Exponents helps
- Homework: review the key idea behind Zero and Negative Exponents before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Zero and Negative Exponents without notes.
- Visual learning: play with the interactive lab to see how Zero and Negative Exponents changes when you adjust its inputs.
- Teaching a friend: explain Zero and Negative Exponents in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Zero and Negative Exponents?
a⁰ = 1 and a⁻ⁿ is the reciprocal of aⁿ.
What will I learn in Zero and Negative Exponents?
Any nonzero base to the power 0 is 1. Negative exponent means flip to the denominator. 2⁻³ = 1/8.
Is Zero and Negative Exponents part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Zero and Negative Exponents lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Zero and Negative Exponents?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.