Laws of indices – from repeated factors to zero, negative and fractional indices

What it shows

Every power is expanded into equal factor tiles: a positive index puts the tiles in the numerator, a negative index puts them in the denominator, and each pair b/b = 1 is crossed out. Counting the factors that are left gives bᵐ·bⁿ = bᵐ⁺ⁿ, bᵐ ÷ bⁿ = bᵐ⁻ⁿ and (bᵐ)ⁿ = bᵐⁿ for any base b ≠ 0, numeric or algebraic. Dividing by b step by step shows why b⁰ = 1 and b⁻ⁿ = 1/bⁿ. Fractional indices follow from (b^(1/n))ⁿ = b, so b^(1/n) = ⁿ√b; the graphs are drawn for x ≥ 0 only.

How to use

On Laws, choose a Law and a Base, then move m and n, including negative values, and count the tiles left after cancelling. On Zero & negative, press Divide by b repeatedly and watch the pattern continue past b⁰. On Fractional, set Numerator m, Denominator n and Base b to see b^(m/n) as a root and a power. On Practice, pick a Question type and press New question. Reset restores the starting values.

Parameters you can change

• Screen Laws, Zero & negative, Fractional, Practice
• Law Multiply: bᵐ · bⁿ, Divide: bᵐ ÷ bⁿ, Power of a power: (bᵐ)ⁿ
• Base 2, 3, 5, 10, x
• Index m -4–6
• Index n -4–6
• Current index (Zero & negative screen) -3–4
• Numerator m of the index m/n -4–6
• Denominator n of the index m/n 1–6
• Base b (Fractional screen) 0–10
• Practice question type All types, Multiply, divide, power, Zero & negative indices, Fractional indices

Questions to explore

1. Why is 2³ · 2⁻³ equal to 1, and what does this tell you about 2⁻³?
2. Why can you not add the indices when you multiply 2³ by 3²?
3. What is 8^(2/3), and why is it easier to take the root first?