Square Roots & Cube Roots

A root is found by splitting the number into prime factors and then grouping them: in pairs for a square root, in threes for a cube root.

The four types asked:

1. Square root of a perfect square — pair the primes, take one from each pair. 1764 = 2²×3²×7² → 2×3×7 = 42.
2. Cube root — group the primes in threes, take one from each group. 3375 = 3³×5³ → 3×5 = 15.
3. Root of a decimal — the root has half as many decimal places as the number. √0.0144 = 0.12, not 0.012.
4. Least multiplier for a perfect square — factorise; whichever prime is left unpaired is the multiplier.

The single most common mistake is treating a root as a division: √1764 is not 1764 ÷ 2. Catch it by squaring your answer back — if 42 × 42 does not return 1764, the answer is wrong.

Speed trick — the last digit. A perfect square ending in 1 has a root ending in 1 or 9; ending in 4 → 2 or 8; in 5 → 5; in 6 → 4 or 6; in 9 → 3 or 7; in 0 → 0. No perfect square ends in 2, 3, 7 or 8. Pair the digits from the right, look at the first pair for the tens digit, and the last digit picks the units.

√: pair the primes · ∛: group in threes · √(a/b) = √a/√b · decimal places halve · check by squaring back

Find √1764. → Factorise: 1764 = 4 × 441 = 2² × 21² = 2² × 3² × 7². Take one from each pair: 2 × 3 × 7 = 42. Sanity check: 42² = 1764, and the number ends in 4 so the root had to end in 2 or 8 — 42 fits.