Laws of Exponents
Laws of Exponents
Exponents are a compact notation for repeated multiplication. Instead of writing 2 × 2 × 2 × 2, you write 2⁴ — shorter, cleaner, and far easier to work with in complicated expressions. The laws of exponents — product, quotient, power, and zero — are the rules that govern how exponential expressions combine.
These laws are not just classroom rules — they are the algebraic backbone of scientific notation, exponential growth and decay, and the logarithm properties you will study later.
This lesson derives each law from the definition of exponentiation, so you understand why the rules work. Once you have that understanding, applying them becomes second nature.
For any nonzero base $a$ and integers $m, n$:
| Law | Rule | Example |
|---|---|---|
| Product | $a^m \cdot a^n = a^{m+n}$ | $x^3 \cdot x^5 = x^8$ |
| Quotient | $\dfrac{a^m}{a^n} = a^{m-n}$ | $\dfrac{x^7}{x^2} = x^5$ |
| Power of Power | $(a^m)^n = a^{mn}$ | $(x^3)^4 = x^{12}$ |
| Power of Product | $(ab)^n = a^n b^n$ | $(2x)^3 = 8x^3$ |
| Power of Quotient | $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ | $\left(\dfrac{x}{3}\right)^2 = \dfrac{x^2}{9}$ |
| Zero Exponent | $a^0 = 1$ | $5^0 = 1$ |
| Negative Exponent | $a^{-n} = \dfrac{1}{a^n}$ | $x^{-3} = \dfrac{1}{x^3}$ |
Simplify $x^4 \cdot x^{-2} \cdot x^3$.
$x^{4 + (-2) + 3} = x^5$.
Simplify $(3x^2y)^3$.
$3^3 \cdot x^{2 \cdot 3} \cdot y^3 = 27x^6 y^3$.
Simplify $\dfrac{3x^4 y^{-2}}{9x^{-1} y^3}$.
$\dfrac{3}{9} \cdot x^{4-(-1)} \cdot y^{-2-3} = \dfrac{1}{3} x^5 y^{-5} = \dfrac{x^5}{3y^5}$.
Simplify $\left(\dfrac{2x^3}{y^2}\right)^{-2}$.
$\dfrac{(2x^3)^{-2}}{(y^2)^{-2}} = \dfrac{y^4}{4x^6}$.
Simplify $\dfrac{(2a^2b)^3}{4a^5b^{-1}}$.
$\dfrac{8a^6b^3}{4a^5b^{-1}} = 2a^{6-5}b^{3-(-1)} = 2ab^4$.
Practice Problems
Show Answer Key
1. $x^{10}$
2. $y^5$
3. $a^{10}$
4. $16x^2y^2$
5. $1$
6. $\dfrac{1}{25}$
7. $\dfrac{x^5}{y^7}$
8. $\dfrac{8}{x^3}$
9. $\dfrac{x^2}{9}$
10. $a^2 b^3$
11. $\dfrac{y^3}{x^2}$
12. $16x^{12}$
13. $3$
14. $\dfrac{x^3 y^6}{z^3}$
15. $\dfrac{1}{x^2}$