Laws of Indices
FoundationThe laws of indices let you simplify powers: same base multiplied means add the indices (aᵐ×aⁿ=aᵐ⁺ⁿ), same base divided means subtract (aᵐ÷aⁿ=aᵐ⁻ⁿ), and a power of a power means multiply ((aᵐ)ⁿ=aᵐⁿ). Remember a⁰=1 and a⁻ⁿ=1/aⁿ and you can handle most questions.
- Same base × → add indices; ÷ → subtract
- Power of a power → multiply indices
- a⁰ = 1 (not 0!)
1Summary 重點摘要
The laws of indices let you simplify powers: same base multiplied means add the indices (aᵐ×aⁿ=aᵐ⁺ⁿ), same base divided means subtract (aᵐ÷aⁿ=aᵐ⁻ⁿ), and a power of a power means multiply ((aᵐ)ⁿ=aᵐⁿ). Remember a⁰=1 and a⁻ⁿ=1/aⁿ and you can handle most questions.
2Vocabulary cards Vocabulary
The number being raised to a power — the a in aⁿ.
The small raised number showing how many times to multiply — the n in aⁿ.
The whole expression aⁿ — read as 'a to the power n'.
1 divided by the number. The reciprocal of a is 1/a, so a⁻ⁿ is the reciprocal of aⁿ.
Another word for index — they mean the same thing.
The index is a positive number — e.g. the 3 in a³.
The index is negative — e.g. a⁻², which becomes 1/a².
The index is 0 — any non-zero number to the power 0 equals 1, e.g. 5⁰ = 1.
3Step by step Steps
- 1. What is a power?Example: a³ = a × a × a
That means 3 a's multiplied together.
Another example: 2⁴ = 2 × 2 × 2 × 2 = 16.
Remember: the small raised number tells you how many times to multiply. - 2. Multiply same bases → add indicesExample: a³ × a⁴ = ?
a³ is 3 a's, a⁴ is 4 a's.
Together that's 3 + 4 = 7 a's.
So a³ × a⁴ = a⁷.
Formula: aᵐ × aⁿ = aᵐ⁺ⁿ - 3. Divide same bases → subtract indicesExample: a⁵ ÷ a² = ?
a⁵ is 5 a's, a² is 2 a's.
When you divide, 2 a's cancel, leaving 5 − 2 = 3.
So a⁵ ÷ a² = a³.
Formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ - 4. Power of a power → multiply indicesExample: (a²)³ = ?
a² means a × a.
(a²)³ means (a × a) multiplied by itself 3 times:
(a × a) × (a × a) × (a × a) = 6 a's total.
So (a²)³ = a⁶.
Formula: (aᵐ)ⁿ = aᵐⁿ - 5. Two special values: a⁰ and negative indicesExample: a⁰ = ?
Think: a⁰ = a⁵ ÷ a⁵ = a⁵⁻⁵ = a⁰.
But a⁵ ÷ a⁵ = 1 (anything divided by itself is 1).
So a⁰ = 1 (for any non-zero number).
Negative index: a⁻² = ?
Think: a⁰ ÷ a² = a⁰⁻² = a⁻².
a⁰ = 1, so a⁻² = 1 ÷ a² = 1/a².
Formula: a⁰ = 1, a⁻ⁿ = 1/aⁿ - 6. What if there are numbers and variables mixed?Example: (2a³b²) × (3a²b⁵) = ?
Step 1: Multiply the numbers first: 2 × 3 = 6.
Step 2: Add indices of a: 3 + 2 = 5 → a⁵.
Step 3: Add indices of b: 2 + 5 = 7 → b⁷.
Answer: 6a⁵b⁷.
Remember: numbers with numbers, a with a, b with b — don't mix them up!
4Diagrams Diagrams
5Worked example Example
Simplify (2a³b²) × (3a²b⁵) ÷ (6ab⁴)
- Step 1: Numbers first: 2 × 3 ÷ 6 = 6 ÷ 6 = 1
- Step 2: Index of a: multiply adds, divide subtracts → 3 + 2 − 1 = 4 → a⁴ (remember a means a¹)
- Step 3: Index of b: 2 + 5 − 4 = 3 → b³
- Step 4: Put it together: 1 × a⁴ × b³ = a⁴b³
- Check with a = 1, b = 1: original (2)(3) ÷ 6 = 1; a⁴b³ = 1 ✓
- Answer: a⁴b³ — add indices when multiplying, subtract when dividing; numbers use ordinary arithmetic.
6Common mistakes Mistakes
7Quick check Quick check
Q1a⁶ × a⁴ = ?
Q2b⁷ ÷ b³ = ?
Q3(x³)⁵ ÷ x⁶ = ?
Q47⁰ = ?
Q52⁻³ = ?
8Recap Recap
- Same base × → add indices; ÷ → subtract
- Power of a power → multiply indices
- a⁰ = 1 (not 0!)
- Negative index = reciprocal (not negative!)
- Numbers with numbers, a with a, b with b