Mathematics · Algebra

Exponent Rules

Reference entry · last updated September 20, 2026

An exponent records repeated multiplication. For a base \(a\) and a positive integer \(n\), \(a^n\) means \(a\) multiplied by itself \(n\) times. Every other exponent rule is a consequence of that definition extended to keep the rules consistent.

1. First principles

For a real base \(a\) and positive integers \(m\) and \(n\), \( a^n = a \cdot a \cdots a \) with \(n\) factors. Multiplying two powers of the same base concatenates the factors, which gives the product rule [1, 2]:

\( a^m \cdot a^n = a^{m+n} \)

2. The core rules

Rule	Statement	Condition
Product	\( a^m a^n = a^{m+n} \)	same base
Quotient	\( \dfrac{a^m}{a^n} = a^{m-n} \)	\( a \neq 0 \)
Power of a power	\( (a^m)^n = a^{mn} \)	all real
Power of a product	\( (ab)^n = a^n b^n \)	all real
Power of a quotient	\( \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \)	\( b \neq 0 \)

3. Zero and negative exponents

The quotient rule with equal exponents gives \( a^0 = a^{n-n} = 1 \) for any \(a \neq 0\). Applying it with a larger denominator exponent gives the negative exponent rule:

\( a^{-n} = \frac{1}{a^n}, \qquad a \neq 0 \)

A negative exponent does not change the sign of the value. It reciprocates the base. \(0^0\) is undefined.

4. Rational exponents and radicals

A rational exponent is defined so that the power rules continue to hold:

\( a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m \)

For even \(n\), \(a\) must be nonnegative to produce a real value. Rational exponents and radical notation describe the same quantity, which links this page to radical operations.

5. Common errors

Exponents do not distribute over addition. \( (a + b)^n \neq a^n + b^n \).
A sum of powers cannot be combined unless the bases and exponents match. \( a^2 + a^3 \) does not simplify to \(a^5\).
\( -a^2 \) means \( -(a^2) \), not \((-a)^2\).
\( 2^{-3} = \dfrac{1}{8} \), not \(-8\).

6. Scientific notation

Scientific notation writes a number as \( a \times 10^n \) with \( 1 \le |a| < 10 \). Multiplying or dividing combines the coefficients and adds or subtracts the exponents, which makes the exponent rules the arithmetic of magnitude.

References

^OpenStax, College Algebra 2e, Rice University (Exponents and scientific notation).
^Michael Sullivan, Algebra and Trigonometry, 11th ed., Pearson, 2019.