Square Roots: How to Calculate Without a Calculator

A square root is the inverse of squaring: the number which, multiplied by itself, gives the number you started with. It is written with the √ symbol. Since 6 × 6 = 36, √36 = 6. It is introduced in Years 5–6 and used constantly at secondary level for the Pythagorean theorem and quadratic equations.

The Square Numbers Worth Memorising

The whole topic becomes easy the moment this table is known. This is not memorising for its own sake — without it, every root question turns into guesswork.

Finding a Square Root Without a Calculator

Method 1: Prime Factorisation

The most reliable route for large numbers.

Example: √576

• 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
• Group into pairs: (2×2)(2×2)(2×2)(3×3)
• Take one from each pair: 2 × 2 × 2 × 3 = 24

Method 2: The Last-Two-Digits Trick

For perfect squares of three or four digits there is a fast route often used in school competitions.

Example: √1,089

• Look at the last two digits: 89. Squares ending in 9 come from 3² = 9 and 7² = 49, so the answer ends in 3 or 7.
• Drop the last two digits, leaving 10. The largest square not exceeding 10 is 3² = 9, so the first digit is 3.
• The candidates are 33 and 37. Since 10 is nearer 9 than 16, take the smaller: 33.
• Check: 33 × 33 = 1,089 ✔

Method 3: Estimating Non-Perfect Squares

√50 lies between √49 = 7 and √64 = 8, and much closer to 7. So √50 ≈ 7.07. Being able to estimate is more useful than computing exactly, because it lets a child spot a wrong calculator answer.

Properties of Roots

• √(a × b) = √a × √b → √36 = √4 × √9 = 2 × 3 = 6
• √(a ÷ b) = √a ÷ √b → √(16/25) = 4/5
• √(a²) = a
• ⚠️ √(a + b) is NOT √a + √b. √(9+16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. This is the costliest error in the topic.

Square Roots of Decimals

√0.25 = 0.5 because 0.5 × 0.5 = 0.25. Note that the root of a number between 0 and 1 is larger than the number itself — counter-intuitive, and something children often reject until it is demonstrated.

The Mistakes That Actually Cost Marks

1. Adding inside the root. Writing √(9+16) = 3 + 4. This is the number one cause and worth testing directly.

2. Thinking √16 = 8. Halving instead of finding what was squared. It signals the square table is not yet known.

3. Forgetting that √1 = 1, not 0.

4. Not realising a root can be simplified. √50 = √(25×2) = 5√2. That form is what secondary exams want, not a decimal.

5. Giving up on large numbers for lack of prime factorisation. Yet 576 and 1,089 can both be done by hand in under a minute.