Laws of Exponents
Formula Sheet 4
1. Product & Quotient Laws
| Law Name | Formula |
|---|---|
| Product Rule Same Base |
$$ x^m \cdot x^n = x^{m+n} $$ |
| Quotient Rule Same Base |
$$ \frac{x^m}{x^n} = x^{m-n} $$ |
| Power of a Power | $$ (x^m)^n = x^{m \cdot n} $$ |
| Power of a Product | $$ (x \cdot y)^m = x^m \cdot y^m $$ |
| Power of a Quotient | $$ \left(\frac{x}{y}\right)^m = \frac{x^m}{y^m} $$ |
2. Special Cases
Zero Exponent
$$ x^0 = 1 $$
(for $$ x \neq 0 $$)
Negative Exponent
$$ x^{-m} = \frac{1}{x^m} $$
(Base moves to denominator)
3. Radical Rules (Roots)
| Rule | Formula |
|---|---|
| Product of Roots |
$$ \sqrt{x} \cdot \sqrt{y} = \sqrt{x \cdot y} $$ (if x, y are non-negative) |
| Fractional Power | $$ x^{\frac{1}{n}} = \sqrt[n]{x} $$ |
| General Root Power | $$ x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m $$ |