Let's break down index notation and its laws. Index Notation is a shorthand way to write repeated multiplication. As we discussed, it has a base and an index (or exponent). For example, xⁿ means x multiplied by itself n times. The Function Law of Index isn't a standard term in mathematics. It's possible you're thinking of the Laws of Indices (also called Laws of Exponents), which are rules for simplifying expressions involving indices. Here are the main Laws of Indices: 1. Product Law: When multiplying expressions with the same base, add the indices. xᵃ × xᵇ = xᵃ⁺ᵇ Example: 2³ × 2² = 2³⁺² = 2⁵ = 32 2. Quotient Law: When dividing expressions with the same base, subtract the indices. xᵃ / xᵇ = xᵃ⁻ᵇ Example: 5⁴ / 5² = 5⁴⁻² = 5² = 25 3. Power of a Power Law: When raising an expression with an index to another power, multiply the indices. (xᵃ)ᵇ = xᵃᵇ Example: (3²)³ = 3²ˣ³ = 3⁶ = 729 4. Zero Index Law: Any non-zero base raised to the power of zero is 1. x⁰ = 1 (where x ≠ 0) Example: 7⁰ = 1 5. Negative Index Law: A base raised to a negative index is equal to the reciprocal of the base raised to the positive index. x⁻ᵃ = 1 / xᵃ Example: 4⁻² = 1 / 4² = 1 / 16 6. Fractional Index Law: A fractional index represents a root. x¹/ⁿ = ⁿ√x xᵐ/ⁿ = (ⁿ√x)ᵐ = ⁿ√(xᵐ) Example: 8²/³ = (³√8)² = 2² = 4 What's next? 📸