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Exponent Rules



Key Terms
Product Rule
Quotient Rule
Power of a Power
Power of a Product
Power of a Quotient
Negative Exponent Rule
Zero Exponent Rule
Rational Exponent Rule
Domain Restrictions
Edge Cases and Common Mistakes
Master Reference of Exponent Laws



The Rules in One Place

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.

Key Terms

The Rules

Product Rule (Exponents)— am⋅an=am+na^m \cdot a^n = a^{m+n}
Quotient Rule (Exponents)— am/an=am−na^m / a^n = a^{m-n}
Power of a Power— (am)n=amn(a^m)^n = a^{mn}
Power of a Product— (ab)n=anbn(ab)^n = a^n b^n
Power of a Quotient— (a/b)n=an/bn(a/b)^n = a^n / b^n
Zero Exponent— a0=1a^0 = 1
Negative Exponent— a−n=1/ana^{-n} = 1/a^n
Rational Exponent— am/n=amna^{m/n} = \sqrt[n]{a^m}

Formulas Used on This Page

Product Rule— am⋅an=am+na^m \cdot a^n = a^{m+n}
Quotient Rule— aman=am−n\frac{a^m}{a^n} = a^{m-n}
Power of a Power— (am)n=amn(a^m)^n = a^{mn}
Power of a Product— (ab)n=an⋅bn(ab)^n = a^n \cdot b^n
Power of a Quotient— (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
Negative Exponent— a−n=1ana^{-n} = \frac{1}{a^n}
Negative Exponent Flip— (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
Zero Exponent— a0=1a^0 = 1

See All Algebra Definitions →

See All Algebra Formulas →


Product Rule

Multiplying two powers that share the same base adds the exponents:

am⋅an=am+na^m \cdot a^n = a^{m+n}


The base stays fixed. Only the exponents combine. The expression x3⋅x5=x8x^3 \cdot x^5 = x^8, and 2−1⋅24=23=82^{-1} \cdot 2^{4} = 2^3 = 8.

The rule requires the bases to match. The product 23⋅342^3 \cdot 3^4 cannot be reduced by adding exponents — different bases mean the rule does not apply.

This identity holds for all real exponents mm and nn, 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:

aman=am−n(a≠0)\frac{a^m}{a^n} = a^{m-n} \qquad (a \neq 0)


The expression x7x2=x5\frac{x^7}{x^2} = x^5, and 5355=5−2=125\frac{5^3}{5^5} = 5^{-2} = \frac{1}{25}.

When m=nm = n, the result is a0=1a^0 = 1. When m<nm < n, the result is a negative exponent — a reciprocal. Both cases are natural consequences of subtracting exponents, and both require a≠0a \neq 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(a^m)^n = a^{m \cdot n}


The expression (x3)4=x12(x^3)^4 = x^{12}, and (2−1)3=2−3=18(2^{-1})^3 = 2^{-3} = \frac{1}{8}.

This rule must not be confused with stacked exponents. The notation amna^{m^n} means a(mn)a^{(m^n)}, evaluated from the top down — it is not the same as (am)n=amn(a^m)^n = a^{mn}. For example, 232=29=5122^{3^2} = 2^9 = 512, while (23)2=26=64(2^3)^2 = 2^6 = 64.

Power of a Product

An exponent applied to a product distributes to each factor:

(ab)n=an⋅bn(ab)^n = a^n \cdot b^n


The expression (3x)4=34⋅x4=81x4(3x)^4 = 3^4 \cdot x^4 = 81x^4. The rule extends to any number of factors: (abc)n=anbncn(abc)^n = a^n b^n c^n.

The distribution works because multiplication is commutative — the factors can be rearranged so that all copies of aa group together and all copies of bb group together.

A critical warning: this rule applies to products only. It does not extend to sums. The expression (a+b)n(a + b)^n is not equal to an+bna^n + b^n. 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:

(ab)n=anbn(b≠0)\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \qquad (b \neq 0)


The expression (25)3=8125\left(\frac{2}{5}\right)^3 = \frac{8}{125}, and (xy2)4=x4y8\left(\frac{x}{y^2}\right)^4 = \frac{x^4}{y^8}.

Combined with the negative exponent rule, this identity also handles cases where the exponent is negative. The expression (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^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=1an(a≠0)a^{-n} = \frac{1}{a^n} \qquad (a \neq 0)


The expression 4−2=1164^{-2} = \frac{1}{16}, and x−1=1xx^{-1} = \frac{1}{x}.

The rule works in both directions. A negative exponent in the denominator moves the term to the numerator: 1a−n=an\frac{1}{a^{-n}} = a^n. An expression like x−3y−2\frac{x^{-3}}{y^{-2}} rewrites as y2x3\frac{y^2}{x^3} — 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 11:

a0=1(a≠0)a^0 = 1 \qquad (a \neq 0)


The result follows from the quotient rule: a0=an−n=anan=1a^0 = a^{n-n} = \frac{a^n}{a^n} = 1.

This holds for every nonzero base — positive, negative, large, small, fractional. The expressions 707^0, (−3)0(-3)^0, and (0.001)0(0.001)^0 all equal 11.

The case a=0a = 0 is excluded. The expression 000^0 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=amn=(an)ma^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m


Both forms are equivalent. The expression 82/3=(83)2=22=48^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4, and 274/3=(273)4=34=8127^{4/3} = (\sqrt[3]{27})^4 = 3^4 = 81.

The special case m=1m = 1 gives the pure root: a1/n=ana^{1/n} = \sqrt[n]{a}. Exponent notation and radical notation are interchangeable representations of the same operation.

When m/nm/n is negative, the negative exponent rule applies first: a−m/n=1am/na^{-m/n} = \frac{1}{a^{m/n}}. 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≥1n \geq 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≠0a \neq 0. Division by zero makes 0−n0^{-n} and 000^0 problematic.

For rational exponents with even roots, the base must be non-negative: a≥0a \geq 0. Even roots of negative numbers are not real.

For irrational exponents and real exponents generally, the base must be strictly positive: a>0a > 0. The continuous extension that defines axa^x for all real xx 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.
Exponent type Restriction on base a Why this restriction
Natural  (n ≥ 1) any real number repeated multiplication is defined for every real a
Zero & negative integer a ≠ 0 0⁰ and 0⁻ⁿ involve division by zero
Rational with even root a ≥ 0 even roots of negatives are not real
Irrational / general real a > 0 continuous extension to ax requires a strictly positive base

Edge Cases and Common Mistakes

Several expressions sit at the boundaries of the rules and require specific attention.

The expression 000^0 is undefined in analysis and equal to 11 by convention in discrete mathematics. It is not covered by the zero exponent rule, which requires a≠0a \neq 0. The zero powers page addresses this case in full.

The expression 0−n0^{-n} is undefined for any positive nn, since it demands division by zero. No law of exponents produces a value for it.

The expressions (−a)n(-a)^n and −an-a^n are not interchangeable. The first raises the negative quantity to the power: (−3)2=9(-3)^2 = 9. The second raises aa to the power and then negates: −32=−9-3^2 = -9. Parentheses determine which interpretation applies.

The most persistent algebraic error is distributing an exponent over a sum: (a+b)n≠an+bn(a + b)^n \neq a^n + b^n. The power of a product rule applies to multiplication, not addition. The expression (2+3)2=25(2 + 3)^2 = 25, while 22+32=132^2 + 3^2 = 13. Expanding (a+b)n(a + b)^n requires the binomial theorem or term-by-term multiplication — the exponent does not distribute.
Expression Interpretation Note
00 undefined in analysis; equal to 1 by convention in discrete math not covered by a0 = 1 (which requires a ≠ 0); see zero powers
0−n  (n > 0) undefined would require dividing by zero
(−a)n  vs  −an (−3)2 = 9  vs  −32 = −9 parentheses change which operation comes first
(a + b)n  ≠  an + bn (2 + 3)2 = 25  vs  22 + 32 = 13 exponent distributes over products, not sums

Master Reference of Exponent Laws

Every rule introduced above belongs to a single family of identities — the laws of exponents. The table below is the master reference: each rule with its formula, a representative example, and the domain restriction that determines when it applies. This is the page in one card, useful both to review and to look up while solving problems elsewhere.
Rule Formula Example Domain restriction
Product am · an = am+n x3 · x5 = x8 bases must match
Quotient am ⁄ an = am−n x7 ⁄ x2 = x5 a ≠ 0
Power of a power (am)n = am·n (x3)4 = x12 distinct from stacked amn
Power of a product (ab)n = an · bn (3x)4 = 81 x4 applies to products, not sums
Power of a quotient (a ⁄ b)n = an ⁄ bn (2 ⁄ 5)3 = 8 ⁄ 125 b ≠ 0
Negative exponent a−n = 1 ⁄ an 4−2 = 1 ⁄ 16 a ≠ 0
Zero exponent a0 = 1 70 = 1,  (−3)0 = 1 a ≠ 0  (00 excluded)
Rational exponent am/n = n√(am) = (n√a)m 82/3 = (∛8)2 = 4 a ≥ 0 when n is even