Negative and Zero Exponents
a⁰ = 1 and a⁻ⁿ is the reciprocal of aⁿ.
Key idea
a⁰ = 1 and a⁻ⁿ is the reciprocal of aⁿ.
In depth
Extending the quotient rule gives a⁰ = 1 (aⁿ/aⁿ = 1) and a⁻ⁿ = 1/aⁿ. A negative exponent means reciprocal, not a negative number.
Worked example
5⁻² = 1/25 = 0.04. And 7⁰ = 1.
Formula
Real-world uses
- Writing very small numbers such as 0.001 as 10⁻³ in science and electronics
Common mistake
Thinking 2⁻³ = −8. It equals 1/8.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Negative and Zero Exponents?
a⁰ = 1 and a⁻ⁿ is the reciprocal of aⁿ. It is a topic at the advanced level in Arithmetic & Number Sense.
What is the formula for Negative and Zero Exponents?
The key formula is a⁻ⁿ = 1/aⁿ.
Where is Negative and Zero Exponents used in real life?
Real-world examples: Writing very small numbers such as 0.001 as 10⁻³ in science and electronics.
What should I learn before Negative and Zero Exponents?
Make sure you are comfortable with Laws of Exponents, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Negative and Zero Exponents?
Thinking 2⁻³ = −8. It equals 1/8.
How can I practise Negative and Zero Exponents?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Negative and Zero Exponents?
Move on to Fractional Exponents next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.