To estimate the square root of a non-perfect square, find the two whole numbers whose squares lie either side of your target. The root lies between those whole numbers; narrow it to one decimal place by testing a middle value. No calculator is needed — just the key perfect squares memorised.
Why do we need to estimate square roots without a calculator?
The KS3 programme of study requires pupils to work with roots and to give approximate values without always relying on a calculator. Exam questions often ask "between which two whole numbers does √n lie?" or ask you to give the value to one decimal place. Knowing how to estimate quickly is also useful for checking whether a calculator answer looks sensible.
What are the key perfect squares to memorise?
Knowing the squares from 1 to 15 gives you the reference points you need for almost all KS3 questions:
| n | n² |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
If a number is not in the right-hand column, its square root is not a whole number.
How do you find which two whole numbers a root lies between?
Find the largest perfect square smaller than your number and the smallest perfect square larger than it. The root lies between those two square numbers.
Example: between which two whole numbers does √50 lie?
- The largest perfect square smaller than 50 is 49 (since 7² = 49).
- The smallest perfect square larger than 50 is 64 (since 8² = 64).
- So 7 < √50 < 8.
Answer: between 7 and 8
How do you narrow down to one decimal place?
Try the value halfway between your two whole numbers, then adjust up or down.
Example: estimate √50 to one decimal place.
- We know 7 < √50 < 8. The midpoint is 7.5.
- 7.5² = 56.25 — that is bigger than 50, so √50 < 7.5.
- Try 7.1: 7.1² = 50.41 — just above 50, so √50 < 7.1.
- Try 7.07: 7.07² = 49.98 — very close, just below 50.
To one decimal place, √50 ≈ 7.1 (since 7.1² = 50.41 is closer to 50 than 7.0² = 49.0).
Worked examples at a glance
| Number | Lower perfect square | Upper perfect square | Root lies between | Estimate to 1 d.p. |
|---|---|---|---|---|
| √20 | 16 (4²) | 25 (5²) | 4 and 5 | 4.5 |
| √50 | 49 (7²) | 64 (8²) | 7 and 8 | 7.1 |
| √90 | 81 (9²) | 100 (10²) | 9 and 10 | 9.5 |
| √130 | 121 (11²) | 144 (12²) | 11 and 12 | 11.4 |
Verify √20: 4.5² = 20.25 ✓ (very close to 20, confirming 4.5 is the 1 d.p. estimate)
Can the same method work for cube roots?
Yes. Replace perfect squares with perfect cubes: 1, 8, 27, 64, 125, 216 (i.e. 1³, 2³, 3³, 4³, 5³, 6³).
Example: estimate ∛60 to one decimal place.
- 3³ = 27 and 4³ = 64, so 3 < ∛60 < 4.
- Try 3.9: 3.9³ = 59.319 — just below 60.
- Try 3.91: 3.91³ ≈ 59.78 — still below.
- Try 3.92: 3.92³ ≈ 60.24 — just above.
To one decimal place, ∛60 ≈ 3.9.
What mistakes do students make?
- Using the wrong pair of perfect squares. For √50, some students write "between 7 and 9" because they confuse 50 with 81. Write the perfect squares out clearly.
- Squaring a decimal incorrectly. 7.1² is not 49.1 or 50.1 — it requires careful multiplication: 7.1 × 7.1 = (7 + 0.1)² = 49 + 1.4 + 0.01 = 50.41.
- Giving the answer as the perfect square root, not the estimate. "√50 lies between 7 and 8" is correct, but "√50 = 7" is wrong.
Frequently asked questions
How many decimal places should I give in an estimate?
Unless the question specifies, one decimal place is standard. A question asking "estimate √50" expects a value like 7.1, not just "between 7 and 8". If it asks "between which two consecutive integers", a range is exactly right.
Why is √2 not a whole number?
Because no whole number multiplied by itself gives exactly 2. The decimal expansion of √2 is 1.41421356… and goes on forever without repeating — √2 is irrational. This is true of any square root of a number that is not a perfect square.
Can I use this method for fourth roots?
Yes. The fourth root of n is the number you multiply by itself four times to get n. Know the fourth powers: 1, 16, 81, 256, 625 (1⁴, 2⁴, 3⁴, 4⁴, 5⁴). For ⁴√100, note that 81 < 100 < 256, so the root lies between 3 and 4. Trying 3.2: 3.2⁴ ≈ 104.86 — so ⁴√100 is just under 3.2.
What is the connection between estimating roots and bounds?
When you say "between 7 and 8", you are establishing the lower and upper bounds for √50. This links directly to GCSE work on error intervals and truncation, where you need to state the range within which a value lies. The estimation method here gives you exactly the kind of bounding argument you will use in bounds problems later.
For Socratic number practice with Professor Pi at KS3, visit aitutors.me.