There are three index rules worth knowing, and each one is just a shortcut for writing the factors out. Learn them by expanding first, and you will stop mixing them up.
What is the small gap?
Many students remember the rules as a chant (“add, subtract, multiply”) without the condition that goes with each. They then use a rule where it does not apply, or pair the wrong operation with the wrong rule.
What do the three rules say, and why?
| Rule | Example | Why it works |
|---|---|---|
| Multiply, same base: add indices | a³ × a⁴ = a⁷ | (a·a·a) × (a·a·a·a) is seven a’s |
| Divide, same base: subtract indices | a⁶ ÷ a² = a⁴ | six a’s over two a’s leaves four |
| Power of a power: multiply indices | (a³)² = a⁶ | (a·a·a) twice is six a’s |
A scaffolded example
An original problem: simplify (2x³)² × x⁴ ÷ x².
Take it in parts. (2x³)² means 2x³ × 2x³, which is 4x⁶ (the 2 is squared to 4, and the power of a power multiplies 3 × 2 = 6).
Then 4x⁶ × x⁴ = 4x¹⁰ (add 6 + 4). Then 4x¹⁰ ÷ x² = 4x⁸ (subtract 10 − 2).
The answer is 4x⁸. Test with x = 1: the original is (2)² × 1 ÷ 1 = 4, and 4x⁸ = 4. Test with x = 2: (16)² × 16 ÷ 4 = 256 × 4 = 1024, and 4 × 256 = 1024.
What does the common mistake look like?
A student writes 2³ × 2⁴ = 4⁷. They multiplied the bases. But the base stays the same: 2³ × 2⁴ = 2⁷ = 128, while 4⁷ = 16 384, nowhere close.
Another slip is ignoring the coefficient in (2x³)². Writing 2x⁶ forgets that the 2 is inside the bracket and is also squared.
Self-check
Simplify 3a² × 4a⁵, then (a⁴)³ ÷ a⁷.
Answer
3a² × 4a⁵: multiply the numbers, 3 × 4 = 12, and add the indices, 2 + 5 = 7, so 12a⁷.
(a⁴)³ = a¹² (multiply 4 × 3). Then a¹² ÷ a⁷ = a⁵ (subtract).
Where does this show up in SPM?
In SPM Mathematics, it is the engine behind algebraic simplification, quadratic work and standard form. The next skill, writing standard form, uses powers of 10 with these rules.
If 2⁵ still looks like 2 × 5, go back to interpreting powers and roots. The hub shows the whole order.
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