To learn indices quickly and correctly, begin with the meaning of a power: it is a compact way to write repeated multiplication of the same factor. Once you understand what the base and index mean, rules for multiplying powers, dividing powers and handling brackets become easier to remember and much less likely to be misused. These eduKateSG Mathematics Quick Notes teach the essential steps, explain the reasons behind them and give you a short independent practice set.
- Read the notation: 2³ means 2 × 2 × 2, which equals 8.
- Identify the base: it is the factor being repeated.
- Multiply powers with the same base: add the indices.
- Divide powers with the same non-zero base: subtract the indices.
- Raise a power to a power: multiply the indices.
- Check: expand a simple expression to see whether the index rule makes sense.
First, separate the base from the index. In 5³, the base is 5 and the index, or exponent, is 3. This means 5 × 5 × 5 = 125. It does not mean 5 × 3. Likewise, 2⁴ = 2 × 2 × 2 × 2 = 16. Say the expression aloud as “five cubed” or “two to the power of four”, then expand it using repeated factors. Spend a moment recognising the structure before memorising any law.
Second, discover the multiplication law by counting equal factors. Consider 2³ × 2². Write it in full: (2 × 2 × 2) × (2 × 2). That is five factors of 2, so the answer is 2⁵ = 32. You can therefore say that aᵐ × aⁿ = aᵐ⁺ⁿ when the bases are the same. The indices are added because two groups of repeated factors have been joined. With different bases, such as 2³ × 3², you cannot simply add the indices; evaluate or use another valid property.
Third, treat division as the reverse of multiplication. In 3⁵ ÷ 3², two factors of 3 can be cancelled from the numerator and denominator, leaving 3³ = 27. For the same non-zero base, the rule is aᵐ ÷ aⁿ = aᵐ⁻ⁿ. If you raise a power to another power, such as (2³)², you have two groups of three factors of 2. There are six factors altogether, so (2³)² = 2⁶ = 64. Here the indices multiply, because the entire repeated product is repeated.
Worked example: use two index laws in one question
Simplify (3² × 3⁴) ÷ 3³. The bases are all 3. Multiply the first two powers by adding indices: 3² × 3⁴ = 3⁶. Divide by 3³ by subtracting indices: 3⁶ ÷ 3³ = 3³. Therefore the answer is 27. A check with ordinary numbers gives (9 × 81) ÷ 27 = 729 ÷ 27 = 27. Both routes agree.
Zero and negative indices: what they really mean
The division law explains why 5⁰ = 1: 5³ ÷ 5³ = 1, while the index rule gives 5³⁻³ = 5⁰. More generally, a⁰ = 1 for non-zero a. Similarly, 2⁻³ = 1/2³ = 1/8. A negative index means a reciprocal, not a negative answer. For non-zero a, a⁻ⁿ = 1/aⁿ. Do not use these rules to conclude that 0⁰ is 1 or that 0 has a negative power; those cases are not defined by the ordinary school rules.
Brackets change what gets raised to the power
Compare (−3)² and −3². The first means (−3) × (−3) = 9. In the second, squaring takes priority over the leading negative sign, giving −(3²) = −9. The brackets decide whether the negative sign belongs to the base. A calculator can give unexpected results if the negative input is not enclosed properly, so write the intended base explicitly.
Useful rules and when they apply
- Same-base multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ.
- Same-base division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ, for a ≠ 0.
- Power of a power: (aᵐ)ⁿ = aᵐⁿ.
- Power of a product: (ab)ⁿ = aⁿbⁿ for integer n where the powers are defined.
- Zero index: a⁰ = 1 when a ≠ 0.
- Negative index: a⁻ⁿ = 1/aⁿ when a ≠ 0.
Mistakes that look tempting
- 2³ = 2 × 3: incorrect; 2³ = 8.
- 2³ + 2² = 2⁵: incorrect; the same-base law is for multiplication, not addition. The sum is 8 + 4 = 12.
- (3²)³ = 3⁵: incorrect; multiply the indices to get 3⁶.
- 5⁻² = −25: incorrect; the value is 1/25.
- Applying a rule across different bases: check the bases before adding or subtracting indices.
Mini-test: one minute to understand, three to solve
- Evaluate 4³.
- Simplify 2⁵ × 2² ÷ 2⁴.
- Simplify (5²)³ and give the answer in index form.
- Find 10⁻² as a fraction.
- Evaluate (−2)⁴ and −2⁴ separately.
Answers and checks
1. 4 × 4 × 4 = 64. 2. 2⁵⁺²⁻⁴ = 2³ = 8. 3. (5²)³ = 5⁶. 4. 10⁻² = 1/100. 5. (−2)⁴ = 16 because an even number of negative factors gives a positive product, whereas −2⁴ = −16. Explain the brackets to someone else to check your understanding.
A six-minute practice plan
Use the first two minutes to expand three small powers without a calculator. Spend the next two using one multiplication law and one division law, then work through a bracketed negative base. In the final minute, check the results by expanding simple examples. Tomorrow, solve a mixed question without notes. If you are stuck, count equal factors again: the laws become easier when their meaning stays visible.
Where to go next
Continue with Square Roots and Cube Roots · Negative Numbers · Algebra · Mathematics. Each Quick Notes article teaches one small topic. Read, try a question without looking, check and repeat tomorrow.
For a longer guide to mathematical foundations, see the immutable reference Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The original page has not been changed.