Exponent Rules — All Laws of Exponents Reference A complete reference of exponent rules — every law of exponents you need for algebra and pre-calculus. Includes rules for products, quotients, powers, zero, negative, and fractional exponents, plus their radical equivalents.
Core Exponent Rules Product of Powers
a m ⋅ a n = a m + n a^m \cdot a^n = a^{m+n} a m ⋅ a n = a m + n Quotient of Powers
a m a n = a m − n ( a ≠ 0 ) \frac{a^m}{a^n} = a^{m-n} \quad(a \ne 0) a n a m = a m − n ( a = 0 ) Power of a Power
( a m ) n = a m n (a^m)^n = a^{m n} ( a m ) n = a mn Power of a Product
( a b ) n = a n b n (ab)^n = a^n b^n ( ab ) n = a n b n Power of a Quotient
( a b ) n = a n b n ( b ≠ 0 ) \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad(b \ne 0) ( b a ) n = b n a n ( b = 0 ) Zero, Negative & One Exponents Zero Exponent
a 0 = 1 ( a ≠ 0 ) a^0 = 1 \quad(a \ne 0) a 0 = 1 ( a = 0 ) Negative Exponent
a − n = 1 a n ( a ≠ 0 ) a^{-n} = \frac{1}{a^n} \quad(a \ne 0) a − n = a n 1 ( a = 0 ) Negative Exponent (Reciprocal)
( a b ) − n = ( b a ) n \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n ( b a ) − n = ( a b ) n Fractional Exponents & Radicals Unit Fractional
a 1 / n = a n a^{1/n} = \sqrt[n]{a} a 1/ n = n a General Fractional
a m / n = a m n = ( a n ) m a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m a m / n = n a m = ( n a ) m Square Root as Exponent
a = a 1 / 2 \sqrt{a} = a^{1/2} a = a 1/2 Cube Root as Exponent
a 3 = a 1 / 3 \sqrt[3]{a} = a^{1/3} 3 a = a 1/3 Negative Fractional
a − m / n = 1 a m n a^{-m/n} = \frac{1}{\sqrt[n]{a^m}} a − m / n = n a m 1 Special Bases Powers of 10
10 n = 100 ⋯ 0 ⏟ n zeros 10^n = \underbrace{100\cdots0}_{n\text{ zeros}} 1 0 n = n zeros 100 ⋯ 0 Scientific Notation
a × 10 n ( 1 ≤ ∣ a ∣ < 10 ) a \times 10^n \quad(1 \le |a| < 10) a × 1 0 n ( 1 ≤ ∣ a ∣ < 10 ) Powers of e (definition)
e x = lim n → ∞ ( 1 + x n ) n e^x = \lim_{n \to \infty}\left(1 + \tfrac{x}{n}\right)^n e x = n → ∞ lim ( 1 + n x ) n Powers of e (series)
e x = ∑ k = 0 ∞ x k k ! e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} e x = k = 0 ∑ ∞ k ! x k Solving Exponential Equations Same Base
a x = a y ⟺ x = y ( a > 0 , a ≠ 1 ) a^x = a^y \iff x = y \quad(a > 0,\ a \ne 1) a x = a y ⟺ x = y ( a > 0 , a = 1 ) Take the Log (any base)
a x = b ⟺ x = log a b a^x = b \iff x = \log_a b a x = b ⟺ x = log a b Change of Base
a x = b ⟺ x = ln b ln a a^x = b \iff x = \frac{\ln b}{\ln a} a x = b ⟺ x = ln a ln b Compound Interest
A = P ( 1 + r n ) n t A = P\left(1 + \tfrac{r}{n}\right)^{n t} A = P ( 1 + n r ) n t 