Square roots

A square has area 49 sq. cm. How long is its side? 7 × 7 = 49, so the side is 7 cm. We call 7 the square root of 49. A square root is squaring done backwards.

In general, if y = x², then x is the square root of y. Squaring and taking the square root undo each other.

One wrinkle. What is the square root of 64? 8 × 8 = 64, so 8. But −8 × −8 = 64 too. Both 8² and (−8)² equal 64.

So every perfect square has two integer square roots, one positive and one negative. With the root sign: √64 = ±8, √100 = ±10, and in general √(n²) = ±n. In this chapter we use only the positive square root.

Given 576 or 327, how can we tell if it's a perfect square at all? The last-digit test rules out 327 at once (squares never end in 7). But 576 ends in 6, which proves nothing. Here are three methods.

1. List the squares until you reach the number or pass it: 20² = 400, 21² = 441, 22² = 484, 23² = 529, 24² = 576. So √576 = 24. It works, but it gets slow for big numbers.

2. Subtract odd numbers in turn. For 81: 81 − 1 = 80, − 3 = 77, − 5 = 72, − 7 = 65, − 9 = 56, − 11 = 45, − 13 = 32, − 15 = 17, − 17 = 0. Zero at the 9th step, so √81 = 9. It works for 729 too, but slowly.

3. Prime factorisation, the best. Split the number into primes. If they can be sorted into two identical groups, one group multiplied out is the square root. Think of it as: each pair inside the root sign comes out as one. 324 = 2 × 2 × 3 × 3 × 3 × 3 = (2 × 3 × 3) × (2 × 3 × 3) = 18², so √324 = 18. Try 156 = 2 × 2 × 3 × 13: the 3 and the 13 have no partners, so 156 is not a perfect square.