Algebra
Exponents and Power Rules
An exponent records repeated multiplication for positive integers and extends consistently to zero, negative, and rational powers. The familiar rules follow from combining equal factors.
Reviewed
Products and quotients with the same base
Multiplying powers of the same base joins their factors, so the exponents add. Dividing cancels factors, so the exponents subtract when the base is nonzero.
These rules require the same base. There is no corresponding rule that turns 2^3 + 2^4 into 2^7 because addition does not combine repeated factors.
a^m a^n = a^(m+n); a^m / a^n = a^(m-n), a != 0Power rules and their conditions
The base must match for the product and quotient rules. Restrictions such as a not equal to zero matter whenever division is involved.
| Situation | Rule | Condition or reason |
|---|---|---|
| Same base, multiply | a^m a^n = a^(m+n) | Concatenate m and n repeated factors |
| Same base, divide | a^m / a^n = a^(m-n) | a must not equal 0 |
| Power of a power | (a^m)^n = a^(mn) | There are n groups of m factors |
| Power of a product | (ab)^n = a^n b^n | Applies to every factor |
| Rational exponent | a^(p/q) = qth-root(a^p) | For real even roots, require a >= 0 |
There is no matching rule for addition: a^m + a^n does not generally equal a^(m+n).
Worked example 1
Simplify (x^3 y^-2)^2
Apply the outer power to each factor.
(x^3)^2 = x^6(y^-2)^2 = y^-4y^-4 = 1/y^4
Result: x^6 / y^4, for y != 0
Powers of powers and products
Raising a power to another integer power repeats the original group, so exponents multiply. A power applied to a product or quotient distributes to every factor.
Parentheses matter: (-3)^2 is 9, while -3^2 means -(3^2) under the standard order of operations.
(a^m)^n = a^(mn); (ab)^n = a^n b^nOperations that look similar but are not exponent rules
- Adding exponents when the bases differ.
- Distributing an exponent over addition: (a + b)^2 is not a^2 + b^2.
- Ignoring parentheses around a negative base.
Worked example 2
Evaluate 27^(2/3)
Take the cube root, then square.
cube-root(27) = 33^2 = 9
Result: 27^(2/3) = 9
Zero and rational exponents
For nonzero a, a^0 = 1 keeps the quotient rule consistent because a^m/a^m = a^0 = 1. A rational exponent combines a root and a power.
In real-number calculations, even roots require a nonnegative radicand. Complex numbers extend the possible values but can introduce multiple branches.
a^(m/n) = (nth-root(a))^mQuestions about exponent rules
Why is 0^0 special?
Different mathematical contexts assign different conventions, so elementary real arithmetic usually treats it as indeterminate rather than applying a rule blindly.
Can fractional exponents produce more than one value?
The real principal value is normally intended in calculator work. Complex exponentiation requires branch conventions.