The natural exponent definition — multiply the base by itself n times — has no way to handle a0 or a−3. There is no such thing as multiplying a number by itself zero times or negative three times. Yet the laws of exponents demand that these expressions have values, and they leave no room for choice about what those values must be.
Motivation: Extending the Pattern
The quotient rule for natural exponents states that anam=am−n when m>n. But the left side of that equation makes sense even when m=n or m<n — the fraction a3a3 and the fraction a5a2 are both perfectly well-defined as long as a=0.
When m=n, the left side gives anan=1, while the right side gives an−n=a0. If the rule is to hold, then a0 must equal 1.
When m<n, the left side produces a fraction. The expression a5a2=a31 by cancellation, while the right side gives a2−5=a−3. If the rule is to hold, then a−3 must equal a31.
Neither definition is a choice. Both are forced by the requirement that the quotient rule — already established for natural exponents — continues to work when the exponent crosses zero into negative territory.
The Zero Exponent
For any nonzero a:
a0=1
The quotient rule provides one proof: a0=an−n=anan=1. But the same conclusion emerges from a purely numerical pattern. The powers of 3 descend by a factor of 3 at each step: 33=27, 32=9, 31=3. Continuing the pattern — dividing by 3 each time — gives 30=1. The same sequence works for any nonzero base: 50=1, (−7)0=1, (0.01)0=1.
The restriction a=0 is essential. The expression 00 sits at the intersection of two conflicting patterns. Following a0=1 suggests 00=1. Following 0n=0 suggests 00=0. Since both patterns have equal claim and produce different answers, 00 is left undefined — or, in certain contexts, assigned a value by convention depending on the application, most commonly 1 in combinatorics and series.
Definition of Negative Exponents
For any nonzero a and any natural number n:
a−n=an1
A negative exponent produces the reciprocal of the corresponding positive power. The expression 2−3=231=81. The expression 5−1=51. The expression 10−4=100001=0.0001.
This definition is not arbitrary — it is the only one compatible with the product rule. The product a3⋅a−3 must equal a3+(−3)=a0=1. For that to hold, a−3 must be the value that multiplies with a3 to give 1 — which is a31.
When the base is itself a fraction, the reciprocal interpretation applies in full. The expression (31)−2 means the reciprocal of (31)2=91, which is 9. Equivalently, (31)−2=32=9. A negative exponent on a fraction flips it before raising to the positive power.
Double Negatives and Reciprocals
The reciprocal interpretation extends cleanly to fractions and nested negatives.
A negative exponent on a fraction inverts it:
(ba)−n=(ab)n
The expression (52)−3=(25)3=8125. The negative exponent flips the fraction, and the positive exponent that remains is applied normally.
The case n=1 gives the simplest form of the reciprocal: a−1=a1. This notation appears throughout algebra and beyond — a−1 is the standard way to write the multiplicative inverse of a.
A negative exponent in the denominator moves the expression to the numerator. The fraction a−n1 equals an, since taking the reciprocal of a reciprocal returns the original. Similarly, b−3a−2=a2b3 — each negative exponent crosses the fraction bar and becomes positive.
Verifying the Laws Still Hold
The definition a−n=an1 was chosen to preserve the laws of exponents. Verification confirms that each rule carries through without exception.
The product rule: a−2⋅a−3 should equal a−2+(−3)=a−5. Computing directly: a21⋅a31=a51=a−5. The rule holds.
Mixed signs: a3⋅a−5 should equal a3+(−5)=a−2. Computing directly: a3⋅a51=a5a3=a21=a−2. The rule holds.
Power of a power: (a−2)3 should equal a(−2)(3)=a−6. Computing directly: (a−2)3=a−2⋅a−2⋅a−2=a21⋅a21⋅a21=a61=a−6. The rule holds.
This is the central point. Negative exponents were not defined by intuition or analogy — they were defined as the unique values that make the algebraic machinery work. Every law that held for natural exponents continues to hold for all integers, because the definition was engineered to guarantee exactly that.
Worked Examples
Simplify 4−2. Applying the definition: 4−2=421=161.
Simplify x7x3. The quotient rule gives x3−7=x−4=x41. Both forms — x−4 and x41 — are correct; context determines which is preferred.
Simplify (2a−3)2. Distribute the exponent: 22⋅(a−3)2=4a−6=a64.
Simplify a4b−1a−2b3. Handle each variable separately. For a: a−2−4=a−6. For b: b3−(−1)=b3+1=b4. The result is a−6b4=a6b4.
Simplify (y2x−1)−3. The negative exponent flips the fraction: (x−1y2)3=(1y2⋅x)3=(xy2)3=x3y6.
In each case, the laws reduce the expression step by step. Negative exponents are not obstacles — they are instructions to take reciprocals, and they simplify by the same rules that govern every other exponent.