What is a square?
The square of a number is the number multiplied by itself. We write it with a small raised 2. For example, 5² = 5 × 5 = 25. We read 5² as "5 squared". A whole number obtained by squaring another whole number is called a perfect square. The first few perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
Key idea / Key idea
n² = n × n. Squaring gives the area of a square with side n. So a square of side 7 cm has area 7² = 49 cm².
What is a square root?
The square root of a number is the value that, when squared, gives that number. The symbol is √. Since 6² = 36, we say √36 = 6. Square root is the reverse (inverse) of squaring. To find a square root, you can use prime factorisation: pair up equal prime factors and take one from each pair.
Worked example / Worked example
Find √324 by prime factorisation.
324 = 2 × 2 × 3 × 3 × 3 × 3
Group in pairs: (2 × 2) × (3 × 3) × (3 × 3)
Take one from each pair: 2 × 3 × 3 = 18.
Check: 18² = 18 × 18 = 324. ✔ So √324 = 18.
You can also estimate a square root by finding the two perfect squares it lies between. For example √50 is between √49 = 7 and √64 = 8, and it is close to 7. Fractions and decimals can be squared too: (0.2)² = 0.04 and (½)² = ¼.
Remember / Remember
- Squaring and square root are inverse operations.
- A perfect square is never negative.
- √(a × b) = √a × √b, so √36 = √4 × √9 = 2 × 3 = 6.
- Read √ as "square root of".