Exponent Rules
Exponent Rules
An exponent tells you how many times to multiply a base by itself: a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}. Working comfortably with exponent rules lets you simplify expressions instantly instead of writing out long products.
Product Rule
When multiplying powers with the same base, add the exponents:
a^m \times a^n = a^{m+n}This makes sense because a^m contributes m copies of a and a^n contributes n more, for m+n copies total. For example, 2^3 \times 2^4 = 2^7 = 128.
Quotient Rule
When dividing powers with the same base, subtract the exponents:
\frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)For example, \frac{5^6}{5^2} = 5^4 = 625.
Power Rule
When raising a power to another power, multiply the exponents:
(a^m)^n = a^{mn}For example, (3^2)^4 = 3^8 = 6561. This is different from a^m \times a^n — be careful not to mix up "power of a power" with "product of powers."
Two related rules distribute an exponent across a product or quotient: (ab)^n = a^n b^n and \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.
Zero Exponent
Any nonzero base raised to the 0 power equals 1:
a^0 = 1 \quad (a \neq 0)This follows from the quotient rule: \frac{a^n}{a^n} = a^{n-n} = a^0, but also \frac{a^n}{a^n} = 1 since anything divided by itself is 1. So a^0 must equal 1.
Negative Exponents
A negative exponent means "take the reciprocal":
a^{-n} = \frac{1}{a^n} \quad (a \neq 0)For instance, 2^{-3} = \frac{1}{2^3} = \frac{1}{8}. This also follows from the quotient rule: \frac{a^2}{a^5} = a^{2-5} = a^{-3}, but direct division gives \frac{a^2}{a^5} = \frac{1}{a^3}, so a^{-3} = \frac{1}{a^3}.
Example
Simplify \frac{2^5 \times 2^{-2}}{2^{-4}}.
Step 1: Combine the numerator using the product rule: 2^5 \times 2^{-2} = 2^{5+(-2)} = 2^3.
Step 2: Now divide using the quotient rule: \frac{2^3}{2^{-4}} = 2^{3-(-4)} = 2^7.
Step 3: Evaluate: 2^7 = 128.
Key Takeaways
- Product rule: a^m \cdot a^n = a^{m+n}. Quotient rule: a^m / a^n = a^{m-n}. Power rule: (a^m)^n = a^{mn}.
- (ab)^n = a^n b^n and (a/b)^n = a^n/b^n distribute exponents over products and quotients.
- Any nonzero number to the 0 power is 1.
- A negative exponent means "reciprocal": a^{-n} = 1/a^n.