01 · EXPONENTIAL RULES
Fundamental Laws of Exponents
Let a > 0 be a real number and m, n be rational numbers:
LAWS SUMMARY
Core Exponential Formulas
1. Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
2. Quotient Rule: aᵐ / aⁿ = aᵐ⁻ⁿ
3. Power Rule: (aᵐ)ⁿ = aᵐⁿ
4. Power of a Product: aᵐ × bᵐ = (ab)ᵐ
5. Zero Exponent: a⁰ = 1 (where a ≠ 0)
6. Negative Exponent: a⁻ᵐ = 1 / aᵐ
02 · RATIONAL EXPONENTS & RADICALS
Fractional Powers (aᵐ/ⁿ)
Let a > 0 be a real number and n a positive integer. Then a¹/ⁿ = ⁿ√a.
SOLVED EXAMPLES
Simplifying Rational Exponents
Example 1: Evaluate 64¹/²
64¹/² = (8²)¹/² = 8² ⁽¹/²⁾ = 8¹ = 8
Example 2: Evaluate 32²/⁵
32²/⁵ = (2⁵)²/⁵ = 2⁵ ⁽²/⁵⁾ = 2² = 4
Example 3: Simplify (2²/³ × 2¹/⁵)
2²/³ ⁺ ¹/⁵ = 2⁽¹⁰ ⁺ ³⁾/¹⁵ = 2¹³/¹⁵