The laws of exponents were derived step by step across the preceding pages — first for natural exponents, then verified as the definition extended to negative, rational, and irrational exponents. This page collects every rule, its domain restrictions, and the edge cases that require caution.
Product Rule
Multiplying two powers that share the same base adds the exponents:
am⋅an=am+n
The base stays fixed. Only the exponents combine. The expression x3⋅x5=x8, and 2−1⋅24=23=8.
The rule requires the bases to match. The product 23⋅34 cannot be reduced by adding exponents — different bases mean the rule does not apply.
This identity holds for all real exponents m and n, subject to the domain restrictions on the base outlined later on this page.
Quotient Rule
Dividing two powers that share the same base subtracts the exponents:
anam=am−n(a=0)
The expression x2x7=x5, and 5553=5−2=251.
When m=n, the result is a0=1. When m<n, the result is a negative exponent — a reciprocal. Both cases are natural consequences of subtracting exponents, and both require a=0.
As with the product rule, the bases must be identical for the rule to apply.
Power of a Power
Raising a power to another power multiplies the exponents:
(am)n=am⋅n
The expression (x3)4=x12, and (2−1)3=2−3=81.
This rule must not be confused with stacked exponents. The notation amn means a(mn), evaluated from the top down — it is not the same as (am)n=amn. For example, 232=29=512, while (23)2=26=64.
Power of a Product
An exponent applied to a product distributes to each factor:
(ab)n=an⋅bn
The expression (3x)4=34⋅x4=81x4. The rule extends to any number of factors: (abc)n=anbncn.
The distribution works because multiplication is commutative — the factors can be rearranged so that all copies of a group together and all copies of b group together.
A critical warning: this rule applies to products only. It does not extend to sums. The expression (a+b)n is not equal to an+bn. Expanding a sum raised to a power requires the binomial theorem or direct multiplication.
Power of a Quotient
An exponent applied to a quotient distributes to numerator and denominator:
(ba)n=bnan(b=0)
The expression (52)3=1258, and (y2x)4=y8x4.
Combined with the negative exponent rule, this identity also handles cases where the exponent is negative. The expression (ba)−n=(ab)n — the negative exponent flips the fraction before the positive power is applied.
Negative Exponent Rule
A negative exponent produces the reciprocal of the corresponding positive power:
a−n=an1(a=0)
The expression 4−2=161, and x−1=x1.
The rule works in both directions. A negative exponent in the denominator moves the term to the numerator: a−n1=an. An expression like y−2x−3 rewrites as x3y2 — each negative exponent crosses the fraction bar and becomes positive.
The full development of this rule, including why the definition is forced by the quotient rule, appears on the negative exponents page.
Zero Exponent Rule
Any nonzero base raised to the power of zero equals 1:
a0=1(a=0)
The result follows from the quotient rule: a0=an−n=anan=1.
This holds for every nonzero base — positive, negative, large, small, fractional. The expressions 70, (−3)0, and (0.001)0 all equal 1.
The case a=0 is excluded. The expression 00 is not covered by this rule — it is addressed separately on the zero powers page.
Rational Exponent Rule
A fractional exponent connects powers to roots:
am/n=nam=(na)m
Both forms are equivalent. The expression 82/3=(38)2=22=4, and 274/3=(327)4=34=81.
The special case m=1 gives the pure root: a1/n=na. Exponent notation and radical notation are interchangeable representations of the same operation.
When m/n is negative, the negative exponent rule applies first: a−m/n=am/n1. The full treatment, including domain considerations for even roots, appears on the rational exponents page.
Domain Restrictions
Each extension of the exponent definition tightens the conditions on the base. The laws themselves do not change — but the set of bases for which they apply narrows.
For natural exponents (n≥1), no restriction exists. Any real number — positive, negative, or zero — can serve as the base.
For zero and negative exponents, the base must be nonzero: a=0. Division by zero makes 0−n and 00 problematic.
For rational exponents with even roots, the base must be non-negative: a≥0. Even roots of negative numbers are not real.
For irrational exponents and real exponents generally, the base must be strictly positive: a>0. The continuous extension that defines ax for all real x requires a positive base to function.
The pattern is monotonic — each stage permits fewer bases than the last. Recognizing which restriction applies to a given expression prevents applying laws outside their valid domain.
Edge Cases and Common Mistakes
Several expressions sit at the boundaries of the rules and require specific attention.
The expression 00 is undefined in analysis and equal to 1 by convention in discrete mathematics. It is not covered by the zero exponent rule, which requires a=0. The zero powers page addresses this case in full.
The expression 0−n is undefined for any positive n, since it demands division by zero. No law of exponents produces a value for it.
The expressions (−a)n and −an are not interchangeable. The first raises the negative quantity to the power: (−3)2=9. The second raises a to the power and then negates: −32=−9. Parentheses determine which interpretation applies.
The most persistent algebraic error is distributing an exponent over a sum: (a+b)n=an+bn. The power of a product rule applies to multiplication, not addition. The expression (2+3)2=25, while 22+32=13. Expanding (a+b)n requires the binomial theorem or term-by-term multiplication — the exponent does not distribute.