A power is repeated multiplication of the same number, and a root undoes it. Holding that one picture makes the later index rules easy to remember.
What is the small gap?
The expression 2⁴ is written with two numbers, so it is easy to treat them like any two numbers: add, or multiply. The index is a counter, not a multiplier. It says how many copies of the base are multiplied together.
How do you read a power and a root?
| Expression | Meaning | Value |
|---|---|---|
| 5² | 5 × 5 | 25 |
| 2⁴ | 2 × 2 × 2 × 2 | 16 |
| 10³ | 10 × 10 × 10 | 1000 |
| √36 | the number that squares to 36 | 6 |
| ∛64 | the number that cubes to 64 | 4 |
A scaffolded example
An original problem: a square tile has sides of 9 cm. A second square has an area of 144 cm². How much longer is the side of the second square?
First tile: side 9 cm. Second tile: the area is side², so side = √144. Ask which number squared gives 144: 12 × 12 = 144, so the side is 12 cm.
The difference is 12 − 9 = 3 cm.
Notice the two operations: squaring goes from side to area, and the square root comes back.
What does the common mistake look like?
A student evaluates 4³ as 4 × 3 = 12. Writing it out exposes the mistake: 4³ = 4 × 4 × 4 = 64.
A second trap is signs. −4² means −(4 × 4) = −16, because the power applies only to the 4. But (−4)² = (−4) × (−4) = 16. The brackets decide what is being squared.
Self-check
Work out 2⁵, then √81, then (−3)³.
Answer
2⁵ = 2 × 2 × 2 × 2 × 2 = 32. √81 = 9, because 9 × 9 = 81.
(−3)³ = (−3) × (−3) × (−3). Two negatives give +9, and +9 × (−3) = −27. An odd number of negative factors gives a negative answer.
Where does this show up in SPM?
In SPM Mathematics, powers appear in algebra, quadratic functions and standard form. In Physics, formulas like v² = u² + 2as and E = ½mv² ask for a square and later a square root.
Next, learn the rules in applying basic index rules. The link to geometry is in Pythagoras’ theorem, and the full sequence is on the powers, roots and scientific notation hub.
One-to-one Mathematics tuition starts with a one-hour trial class (from RM50).