Concept explainerCalculus I · Algebra IFoundational

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.

LaTeX article Updated July 13, 2026

a^ma^n=a^{m+n}\qquad(a^m)^n=a^{mn}

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.

x^3x^5=x^{3+5}=x^8

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.

a^{-3}=\frac1{a^3},\quad a\ne0

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.

(ab)^n=a^nb^n\qquad(a+b)^2\ne a^2+b^2

Worked example

Common mistakes

Adding exponents when adding terms.
Distributing a power across a sum.
Leaving a negative exponent as though it means a negative value.

Keep these ideas

Exponent rules describe factor counts.
Negative powers indicate reciprocals.
Products and sums behave differently.