A power records repeated multiplication, not multiplication of the base by the index. This Secondary 1 Mathematics guide for Sembawang families begins with that distinction, then explains when common index laws apply.
For example, 3⁴ means 3 × 3 × 3 × 3 = 81. It does not mean 3 × 4. Expanding a small power into its factors is a useful way to check whether a rule has been understood.
Continue your Secondary 1 Mathematics journey: Bukit Timah Secondary 1 Mathematics programme · Parent reading guide · Number foundations. For the 3-pax teaching approach, class fit and current arrangements, begin with the programme page.
Identify the base and the index · Add indices when multiplying powers of the same base · Subtract indices when dividing matching powers · Multiply indices for a power of a power · Use brackets to show a negative base · Explain the zero index when it is introduced · Try a short notation check · Use a precise example to discuss support
Identify the base and the index
In 5³, the base is five and the index is three. There are three factors, each equal to five, so the value is 125.
Compare 5³ with 3⁵. The latter contains five factors of three and equals 243. Swapping the base and index generally changes the value.
A student who confuses powers with multiplication may need only two or three factor expansions before using a rule. The aim is to make the notation readable.
Add indices when multiplying powers of the same base
Consider 2³ × 2². The first factor contributes three copies of two and the second contributes two more. There are five copies altogether, so the product is 2⁵ = 32.
The rule aᵐ × aⁿ = aᵐ⁺ⁿ describes this combination of factors. In the positive-integer examples here, the shared base makes that explanation direct.
It does not turn 2³ + 2² into 2⁵. The sum is eight plus four, which equals twelve. Check the operation sign before using an index law.
Subtract indices when dividing matching powers
For 3⁵ ÷ 3², write the quotient as five factors of three divided by two factors of three. Cancelling the two shared factors leaves 3³ = 27.
This illustrates aᵐ ÷ aⁿ = aᵐ⁻ⁿ for a non-zero base, with the index ranges used in the student’s course. The base cannot be zero because the denominator would be zero.
If division has not yet been connected to negative indices, start with a larger numerator index so the remaining factors are visible. More advanced cases can wait until they are part of the school topic.
Multiply indices for a power of a power
In (2³)², the bracketed quantity 2³ is multiplied by itself: 2³ × 2³ = 2⁶ = 64.
The outer power repeats the entire inner product. Thus (aᵐ)ⁿ = aᵐⁿ in these positive-integer examples. Adding the indices would count the factors incorrectly.
Contrast (2³)² with 2³ × 2². The first gives 2⁶, while the second gives 2⁵. Similar-looking symbols can describe different structures.
Use brackets to show a negative base
The expression (−3)² has negative three as its base and equals (−3)(−3) = 9.
In −3², the power is evaluated before the leading minus, giving −(3 × 3) = −9. The brackets therefore matter.
For odd powers, (−3)³ = −27 because multiplying three negative factors produces a negative result. Count the factors or signs when needed rather than assuming every power makes a value positive.
Explain the zero index when it is introduced
For a non-zero base, a³ ÷ a³ = 1 because a non-zero quantity divided by itself is one. The index rule also writes this as a⁰. This explains the convention a⁰ = 1 for non-zero a.
Keep that condition visible. The same division argument cannot be used for a = 0 because it would require dividing by zero.
Use this extension only alongside the student’s current learning. Secure meaning for positive powers is a sensible starting point before zero or negative indices.
Try a short notation check
- Evaluate 4³.
- Write 5² × 5³ as one power of five.
- Write 7⁴ ÷ 7² as one power of seven.
- Write (3²)³ as one power of three.
- Evaluate (−2)⁴ and −2⁴.
Answers: 64; 5⁵; 7²; 3⁶; and 16 versus −16. In the last pair, identify the base before calculating.
Ask for one expansion into factors as well as the answer. If the expansion is correct but the final number is wrong, the next repair is arithmetic rather than notation.
Use a precise example to discuss support
For Sembawang families, useful evidence is a pair of expressions the student treats as interchangeable, such as a sum of powers and a product of powers. That comparison makes the misconception easier to discuss.
The Bukit Timah programme page explains the 3-pax approach and provides the route for checking current arrangements. Follow the student’s actual topic sequence rather than introducing every index rule at once.
Choose the next useful reading step
Number foundations · PSLE to algebra · Diagnose learning gaps
For a conversation about suitable support, continue to Secondary 1 Mathematics tuition in Bukit Timah. Bring recent marked work and the student’s current school topic list so the starting point is clear.