Exponent rules that come from counting factors
The rules are compressed descriptions of repeated multiplication. Knowing where they come from prevents powers from spreading across sums illegally.
What the idea means, why its conditions matter, and where it connects.
Updated July 13, 2026Same-base multiplication joins factor lists, division removes factors, and a power of a power repeats the whole list.
Products add exponents
When bases match, multiplying combines repeated copies of the same factor. The total number of copies is the sum of the exponents.
The bases must match. There is no comparable rule for x² times y³ because the repeated factors are different.
Quotients subtract and explain negative powers
Dividing equal bases cancels common factors. If more factors remain in the denominator, a negative exponent records their reciprocal location.
A zero exponent is the balanced case: every nonzero base divided by itself equals one.
Powers distribute over products, not sums
Raising a product to a power repeats every factor, so each factor receives the exponent. A sum is not a factor list and cannot be handled term by term.
Expanding (a + b)² shows the missing middle term that a false distribution would erase.
Simplify (3x²y⁻¹)² · x³y.
Apply the outer power to each factor.
Add exponents for matching bases.
Write the final answer with positive exponents.
Common mistakes
- Adding exponents when adding terms.
- Distributing a power across a sum.
- Leaving a negative exponent as though it means a negative value.
Three takeaways
- Exponent rules describe factor counts.
- Negative powers indicate reciprocals.
- Products and sums behave differently.