1. What an exponent is
The notation $a^n$, with a small raised number, is a shortcut for multiplying $a$ by itself $n$ times:
$$ a^n = \underbrace{a \cdot a \cdot a \cdot \ldots \cdot a}_{n \text{ copies}}. $$So $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$. The number on the bottom ($a$) is the base, and the one raised up ($n$) is the exponent (or power, or index). $a^n$ as a whole is "$a$ to the $n$th power."
For a number $a$ and a positive integer $n$, $a^n$ means $a$ multiplied by itself $n$ times. Just as multiplication compresses repeated addition, exponentiation compresses repeated multiplication.
Two special powers have their own names: $a^2$ is "$a$ squared" (because it's the area of a square of side $a$); $a^3$ is "$a$ cubed" (the volume of a cube of side $a$). Past three, we just say "$a$ to the fourth," "$a$ to the fifth," and so on.
2. Special cases: $a^0$ and $a^1$
The "repeated multiplication" definition is awkward when $n$ is $0$ or $1$. "Multiplying $a$ by itself zero times" doesn't quite parse. So we define those cases — not arbitrarily, but in the way that's forced if we want the laws of exponents to keep working.
- $a^1 = a$. Multiplying $a$ by itself "once" just gives $a$. Easy.
- $a^0 = 1$ for any nonzero $a$. Less obvious, but consistent. Look at the pattern: $a^3 \to a^2 \to a^1$, each step divides by $a$. Continuing one more step: $a^0$ should be $a^1 \div a = a/a = 1$.
The pattern argument is the friendliest one. Take powers of $2$:
$$ \ldots,\;\; 2^4 = 16,\;\; 2^3 = 8,\;\; 2^2 = 4,\;\; 2^1 = 2,\;\; 2^0 = 1,\;\; 2^{-1} = \tfrac{1}{2},\;\; 2^{-2} = \tfrac{1}{4},\;\; \ldots $$Each step right divides by $2$. For the pattern not to break at $n = 0$ and $n = -1$, we must define $2^0 = 1$ and $2^{-1} = \tfrac{1}{2}$.
The rule "any nonzero number to the zero is $1$" carries an explicit exception. $0^0$ is undefined — the pattern argument breaks because you'd be dividing by $0$, and other arguments give conflicting answers ($0^n = 0$ for any positive $n$ would suggest $0^0 = 0$). Different fields treat $0^0$ differently for convenience, but it's not a fixed value.
3. Negative exponents
The pattern above also tells you what a negative exponent should mean. Each step right in the sequence divides by the base, so:
$$ a^{-n} = \frac{1}{a^n}. $$A negative exponent flips the base into the denominator. $2^{-3} = \tfrac{1}{2^3} = \tfrac{1}{8}$. $5^{-1} = \tfrac{1}{5}$. $a^{-1}$ is just $\tfrac{1}{a}$, the reciprocal.
The negative sign in the exponent has nothing to do with the sign of the resulting number — it indicates "in the denominator" rather than "subtract." $3^{-2} = \tfrac{1}{9}$ is a small positive number, not a negative one.
A fraction with a negative exponent is the same as the reciprocal with a positive exponent: $\left(\tfrac{2}{3}\right)^{-2} = \left(\tfrac{3}{2}\right)^2 = \tfrac{9}{4}$. Flipping the base and changing the sign of the exponent are the same move.
4. The laws of exponents
The handful of rules below are the entire arithmetic of exponents. Memorize them only if you have to — once you see the bookkeeping argument in the next section, you'll be able to recover each one on demand.
| Name | Rule | Example |
|---|---|---|
| Product | $a^m \cdot a^n = a^{m+n}$ | $2^3 \cdot 2^4 = 2^7$ |
| Quotient | $\dfrac{a^m}{a^n} = a^{m-n}$ | $\dfrac{5^7}{5^4} = 5^3$ |
| Power of a power | $(a^m)^n = a^{mn}$ | $(3^2)^4 = 3^8$ |
| Power of a product | $(ab)^n = a^n \cdot b^n$ | $(2 \cdot 5)^3 = 2^3 \cdot 5^3$ |
| Power of a quotient | $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ | $\left(\dfrac{3}{4}\right)^2 = \dfrac{9}{16}$ |
And the two special cases from earlier:
$$ a^0 = 1 \;(a \neq 0), \qquad a^{-n} = \frac{1}{a^n}. $$5. Reading the laws as bookkeeping
The laws aren't five independent facts. They all come from counting how many copies of the base appear once you fully expand the expression. Show this once to yourself and you won't need to memorize the rules again.
Product
$$ a^3 \cdot a^4 = (a \cdot a \cdot a) \cdot (a \cdot a \cdot a \cdot a) = a^{3+4} = a^7. $$Three copies times four copies makes seven copies. So multiplying powers of the same base adds the exponents.
Quotient
$$ \frac{a^5}{a^2} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a} = a \cdot a \cdot a = a^3. $$The two factors of $a$ in the denominator cancel two of the five upstairs, leaving three. Division subtracts the exponents.
Power of a power
$$ (a^3)^4 = a^3 \cdot a^3 \cdot a^3 \cdot a^3 = a^{3+3+3+3} = a^{12}. $$Four groups of three is twelve. Power-of-a-power multiplies the exponents.
Power of a product / quotient
$$ (ab)^3 = ab \cdot ab \cdot ab = a \cdot a \cdot a \cdot b \cdot b \cdot b = a^3 b^3. $$You can rearrange the factors freely (multiplication is commutative and associative), so each base accumulates the same exponent. Same idea for quotients.
The laws of exponents are not memorization — they are counting.