An exponent tells you how many times to multiply a number by itself. Instead of writing \(2 \times 2 \times 2 \times 2 \times 2\), you write \(2^5\) — base 2, exponent 5. Compact, elegant, and powerful.
The base is the number being multiplied. The exponent (or power) is how many times. Together they form an expression like \(a^n\), read as "a to the power of n" or "a to the nth."
Exponents aren't limited to positive integers. You can have zero, negative, and fractional exponents — and each one has a precise, consistent meaning derived from the exponent rules.
Fold a piece of paper in half once — 2 layers. Fold again — 4 layers. Again — 8. After \(n\) folds you have \(2^n\) layers. After just 42 folds, the stack would reach the Moon. That's exponential growth — and it starts with understanding exponents.
Bacterial doubling, compound interest, radioactive decay, signal strength, earthquake magnitude, population growth — all described by exponential functions \(a^x\). Mastering exponents unlocks the language of growth and decay throughout science and finance.
| Zero exponent | \(a^0 = 1\) for any \(a \neq 0\) |
| Negative exponent | \(a^{-n} = \dfrac{1}{a^n}\) |
| Fractional exponent | \(a^{1/n} = \sqrt[n]{a}\) |
| Rational exponent | \(a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\) |
| Exponent of 1 | \(a^1 = a\) |
| Base of 1 | \(1^n = 1\) for any \(n\) |
| Product rule | \(a^m \cdot a^n = a^{m+n}\) |
| Quotient rule | \(\dfrac{a^m}{a^n} = a^{m-n}\) |
| Power of a power | \((a^m)^n = a^{mn}\) |
| Power of a product | \((ab)^n = a^n b^n\) |
| Power of a quotient | \(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\) |
| x | Linear: 2x | Quadratic: x² | Exponential: 2ˣ |
|---|---|---|---|
| 1 | 2 | 1 | 2 |
| 5 | 10 | 25 | 32 |
| 10 | 20 | 100 | 1,024 |
| 20 | 40 | 400 | 1,048,576 |
| 30 | 60 | 900 | 1,073,741,824 |
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Exponent rules appear throughout every level of BUders quizzes.
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