Zero and Negative Integral Indices

1Summary 重點摘要

Learn what a⁰ and a⁻ⁿ mean, know any number to power 0 equals 1.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Definition of zero indexThe zero index states that any non-zero number a raised to the power 0 equals 1, that is a⁰ = 1 (a ≠ 0). Why? By the index laws, aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and any number divided by itself equals 1, so a⁰ must be 1. For example 5⁰ = 1, (−7)⁰ = 1 and (0.3)⁰ = 1. A common mistake is thinking a power of 0 makes the number 0; the result is always 1, but note that 0⁰ is undefined and cannot be substituted.
2. Step 2: Definition of negative indexThe negative index states that a⁻ⁿ = 1/aⁿ (a ≠ 0), meaning a negative index turns the base into its reciprocal with a positive index. For example 2⁻³ = 12³ = 18 and 5⁻² = 15² = 125. This definition is forced by the index laws: aⁿ × a⁻ⁿ = a⁰ = 1, so the two must be reciprocals. A common mistake is reading the negative sign as a negative value, writing 2⁻³ = −8; the sign means reciprocal, and the result stays positive.
3. Step 3: Where the formula comes fromThe negative index formula also follows from the index laws: a⁰ ÷ aⁿ = 1 ÷ aⁿ = 1/aⁿ, while the laws give a⁰⁻ⁿ = a⁻ⁿ; both results must agree, so a⁻ⁿ = 1/aⁿ. The rule is not a new fact but a necessary consequence of the index laws. For example 3⁻² = 13² = 19. A common mistake is memorising the rule without its origin and failing on fractional bases such as (23)⁻²; simply flip the fraction to get (32)² = 94.
4. Step 4: Worked examplesApply the definitions to typical cases: 5⁰ = 1 because any non-zero base to power zero is 1; 3⁻² = 13² = 19 by the reciprocal rule; (12)⁻¹ = 2 because a fraction with a negative index flips upside down, giving 21 = 2; (−2)⁻² = 1/(−2)² = 14, where the base is squared before taking the reciprocal. Common mistakes: forgetting to flip the fraction in (12)⁻¹, or misreading (−2)⁻² as −4; the correct answer 14 must be positive.
5. Step 5: Simplification techniqueWhen simplifying, rewrite each negative index as a reciprocal with a positive index, replace each zero index with 1, then compute in the usual order of operations. For example 2⁰ × 3⁻² = 1 × 19 = 19, and (2⁻¹)⁴ = 2⁻⁴ = 116 by multiplying indices first. Why handle term by term? Indices with different bases cannot be merged directly. A common mistake is mixing up the order, wrongly treating 2⁰ × 3⁻² as (2 × 3)⁻² = 136; process each base separately before multiplying.
6. Step 6: Important cautionsTwo cases need special care: first, 0⁰ is undefined and must not be written as 1; second, a negative index with base 0 is also undefined, because a⁻ⁿ = 1/aⁿ would divide by zero, for example 0⁻² = 10² = 10 is meaningless. For every other positive, negative or fractional base the rules apply normally. Why stress this? Examinations often set traps with a zero base. A common mistake is applying the formula without checking; confirm the base is not 0 before answering.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• a⁰ = 1 (any non-zero number)
• a⁻ⁿ = 1/aⁿ
• Negative index = reciprocal
• Zero index = 1
• 0⁰ is undefined