Exponent Rules
Laws of exponents let you simplify products, quotients and powers.
About this topic
Exponent rules save work when simplifying powers. Multiplying like bases adds the exponents (xᵃ · xᵇ = xᵃ⁺ᵇ), dividing subtracts them (xᵃ / xᵇ = xᵃ⁻ᵇ), and raising a power to a power multiplies them ((xᵃ)ᵇ = xᵃᵇ). A power of a product distributes: (xy)ᵃ = xᵃyᵃ.
These rules come from what an exponent means, repeated multiplication. For example x³ · x⁴ is (x · x · x)(x · x · x · x), which is seven factors of x, so x⁷. Likewise (2x³)² = 2² · x⁶ = 4x⁶, because both the 2 and the x³ get squared. Working through a few of these by writing out the factors makes the rules feel natural rather than memorized.
Exponents are used to express huge and tiny quantities in scientific notation, to compute areas and volumes, and to model growth. Computer memory sizes are powers of 2, and the rules tell you instantly that 2¹⁰ · 2¹⁰ = 2²⁰, about a million.
The biggest trap is applying the product rule to a sum: (x + y)² is not x² + y². Another is adding exponents when the bases differ, such as 2³ · 3². Only like bases combine, and coefficients multiply separately from exponents.
Key ideas
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
Quick check
Use cases: where Exponent Rules helps
- Homework: review the key idea behind Exponent Rules before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Exponent Rules without notes.
- Visual learning: play with the interactive lab to see how Exponent Rules changes when you adjust its inputs.
- Teaching a friend: explain Exponent Rules 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 Exponent Rules?
Laws of exponents let you simplify products, quotients and powers.
What will I learn in Exponent Rules?
aᵐ × aⁿ = aᵐ⁺ⁿ. aᵐ ÷ aⁿ = aᵐ⁻ⁿ. (aᵐ)ⁿ = aᵐⁿ.
Is Exponent Rules 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 Exponent Rules 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 Exponent Rules?
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.