Before powers can be extended to negative, fractional, or irrational exponents, they must first be grounded in the simplest case: a positive whole number telling you how many times to multiply a base by itself. This is where the notation an acquires its first meaning, where the laws of exponents are first derived from concrete arithmetic, and where the patterns emerge that every later extension is built to preserve.
Definition
For a natural number n≥1, the expression an means the product of n copies of a:
an=n timesa⋅a⋅a⋯a
The base a can be any real number — positive, negative, or zero. The exponent n counts the repetitions.
Concrete examples anchor the definition. 24=2⋅2⋅2⋅2=16. 53=5⋅5⋅5=125. (−3)2=(−3)(−3)=9. (−3)3=(−3)(−3)(−3)=−27.
Two special cases follow immediately. When n=1, there is only one copy of the base: a1=a for any a. When the base is 1, repeated multiplication changes nothing: 1n=1 for any n.
A note on convention: whether the natural numbers include 0 varies by source. In this section, natural exponents start at n=1. The case a0 arises naturally from the quotient rule and is addressed on the negative exponents page, where the extension below n=1 is developed.
Sign Behavior
The sign of an depends on the sign of the base and on whether the exponent is even or odd.
A positive base always produces a positive result, regardless of the exponent. 32=9, 35=243, 3100 is positive — no power of a positive number can be negative.
A negative base alternates. When n is even, the negative signs pair off and cancel: (−2)4=(−2)(−2)(−2)(−2)=16. When n is odd, one negative sign remains unpaired: (−2)5=(−2)(−2)(−2)(−2)(−2)=−32. The rule is clean — even exponent yields positive, odd exponent yields negative.
Parentheses determine what counts as the base. The expression (−3)2 squares the entire quantity −3, giving (−3)(−3)=9. The expression −32 squares 3 first and then negates: −(32)=−9. These are not the same number. The exponent binds to the nearest base, so without parentheses, only 3 is raised to the power and the negative sign operates on the result.
The Product Rule
Multiplying two powers of the same base amounts to counting the total number of factors. The expression 23⋅24 writes out as (2⋅2⋅2)(2⋅2⋅2⋅2), which is 7 copies of 2 multiplied together: 27.
The pattern generalizes to any base:
am⋅an=am+n
Three factors of a followed by four more gives seven total — the exponents add. This holds for any natural numbers m and n and any base a.
The bases must match for the rule to apply. The product 23⋅34 cannot be simplified by adding exponents, because the bases are different. No law of exponents combines 23⋅34 into a single power — the expression stays as it is unless rewritten another way.
The Quotient Rule
Dividing two powers of the same base cancels common factors. The expression 2325 writes out as 2⋅2⋅22⋅2⋅2⋅2⋅2. Three copies of 2 cancel between numerator and denominator, leaving 2⋅2=22.
The general rule subtracts exponents:
anam=am−n(m>n,a=0)
Five factors in the numerator minus three in the denominator leaves two — the exponent of the result is the difference.
For natural exponents, the rule requires m>n to keep the result within the natural number framework. When m=n, the expression becomes anan=1, which would correspond to a0. That case lies outside the scope of natural exponents and is the first signal that the definition needs extending — a thread picked up on the negative exponents page.
Power of a Power
Raising a power to another power multiplies the exponents. The expression (23)2 means 23⋅23, which by the product rule equals 23+3=26.
The general rule follows the same logic:
(am)n=am⋅n
The outer exponent n creates n copies of am. The product rule then adds m to itself n times, producing m⋅n.
This rule applies regardless of the values of m and n — as long as both are natural numbers, the result is straightforward multiplication of exponents. The expression (x4)3=x12, and (52)5=510.
Care is needed with notation. The expression amn is not the same as (am)n. Stacked exponents evaluate from the top down: amn=a(mn), not amn. For instance, 232=29=512, while (23)2=26=64.
Power of a Product
An exponent applied to a product distributes to each factor individually. The expression (2⋅3)4 can be expanded as (2⋅3)(2⋅3)(2⋅3)(2⋅3). Rearranging the factors groups all the 2s and all the 3s together: (2⋅2⋅2⋅2)(3⋅3⋅3⋅3)=24⋅34.
The general rule:
(ab)n=an⋅bn
Each factor in the product acquires the exponent independently. This extends to any number of factors: (abc)n=anbncn.
The rule works because multiplication is commutative and associative — the order in which factors are grouped does not affect the product. Rewriting (ab)(ab)(ab) as (a⋅a⋅a)(b⋅b⋅b) is valid precisely because factors can be rearranged freely.
Power of a Quotient
An exponent applied to a quotient distributes to numerator and denominator separately. The expression (32)4 expands as 32⋅32⋅32⋅32=3424=8116.
The general rule:
(ba)n=bnan(b=0)
The logic mirrors the power of a product rule. Multiplying n copies of ba means multiplying n copies of a in the numerator and n copies of b in the denominator. The restriction b=0 is inherited from the requirement that the original fraction be defined.
Combined with the other rules, the power of a quotient enables simplification of complex fractional expressions. The expression (y3x2)4=y12x8, applying both the power of a quotient and the power of a power rule in a single step.
Worked Examples
The laws work together in practice. Simplifying expressions typically requires recognizing which rule applies at each step and applying them in sequence.
Simplify 32⋅35. The bases match, so the product rule gives 32+5=37=2187.
Simplify x3x8. The quotient rule gives x8−3=x5.
Simplify (2x3)4. The power of a product distributes: 24⋅(x3)4=16x12.
Simplify a4b2(a2b)3. Start with the numerator: (a2b)3=a6b3. Then apply the quotient rule to each variable: a4b2a6b3=a6−4b3−2=a2b.
Simplify (y3x2)3⋅y4. The power of a quotient gives y333x6⋅y4=y327x6⋅y4=27x6y.
Each step applies one law. When multiple laws are needed, working from the innermost grouping outward — simplifying parentheses first, then combining like bases — produces the cleanest path to the result.