Prime factorisation cracks open any perfect square or cube exactly — and even when a number is not perfect, you can still estimate its root to one decimal place in seconds.
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Every perfect square breaks into pairs of identical prime factors; every perfect cube breaks into triples.
📌 Finding a root by factorisation
Break the number into prime factors. 1764 = 2 × 2 × 3 × 3 × 7 × 7.
Group into pairs (for square root) or triples (for cube root). 1764 = (2×3×7)² = 42².
Take one factor from each group — √1764 = 2 × 3 × 7 = 42.
✨ Cube root example
2197 in prime factors
13 × 13 × 13
2197 = 13³
Cube root
one factor from the triple
∛2197 = 13
When a number is not a perfect square, you can still estimate its root by finding the two perfect squares it sits between.
Estimation method: √150 sits between 12²=144 and 13²=169. Since 150 is much closer to 144, √150 is close to 12, roughly 12.2.
📊 Roots of decimals
| Number | Root | Why |
|---|---|---|
| √0.0049 | 0.07 | Because 0.07² = 0.0049 |
| √3.24 | 1.8 | Because 1.8² = 3.24 |
| ∛0.008 | 0.2 | Because 0.2³ = 0.008 |
The smallest number to multiply or divide by, to turn a number into a perfect square, comes from the leftover unpaired prime factor.
Trap: 72 = 2³ × 3². The "3²" is already paired, but "2³" leaves one unpaired 2. Multiplying by 2 gives 2⁴ × 3² = a perfect square (144). So the smallest multiplier is 2, not 3.
12 questions on square roots, cube roots and estimation.
