√252 at a glance
- Exact value
- 6√7
- Decimal (10 places)
- 15.8745078664
- Rounded
- 15.9 · 15.87 · 15.875
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.874508
- Prime factorization
- 2² × 3² × 7
- Cube root
- 6.316360
How to simplify √252
Look for the largest perfect square that divides 252. Here it is 36 (6²), because 252 = 36 × 7 and 7 has no square factor left:
The prime factorization tells the same story: 252 = 2² × 3² × 7. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 7 stays inside.
252 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √252 = 2√63, and √63 can be simplified again. Using 36 straight away finishes in one step.
Check: (6√7)² = 6² × 7 = 36 × 7 = 252. As a decimal, 6√7 = 6 × 2.6457513111 ≈ 15.8745078664.
Where √252 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √252 lies between 15 and 16. 252 is 27 above 225 and 4 below 256, so the root is closer to 16.
- Straight line between 225 and 256: 15.8710 (0.02% low)
- Tangent from 15, i.e. 15 + 27 ÷ 30: 15.9000 (0.16% high)
- Tangent from 16, i.e. 16 − 4 ÷ 32: 15.8750 (0% high)
For √252 the tangent at 16 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 252 is just 4 below 256.
Finding √252 with the Babylonian method
Picture a rectangle with an area of 252 and one side x; the other side must be 252 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √252.
Start from the nearest whole number, 16 (16² = 256):
| Step | Guess x | 252 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 16.0000000000 | 15.7500000000 | 15.8750000000 | 3 |
| 2 | 15.8750000000 | 15.8740157480 | 15.8745078740 | 8 |
| 3 | 15.8745078740 | 15.8745078588 | 15.8745078664 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √252 = 15.8745078664 to every decimal shown.
√252 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √252 the pattern is [15; 1, 6, 1, 30] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √252 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 15/1 | 15.0000000000 | 8.7 × 10⁻¹ |
| 16/1 | 16.0000000000 | 1.3 × 10⁻¹ |
| 111/7 | 15.8571428571 | 1.7 × 10⁻² |
| 127/8 | 15.8750000000 | 4.9 × 10⁻⁴ |
| 3,921/247 | 15.8744939271 | 1.4 × 10⁻⁵ |
| 4,048/255 | 15.8745098039 | 1.9 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 252y² = 1. Its smallest solution in positive whole numbers is x = 127, y = 8.
√252 in geometry and everyday measurements
- A square patio or deck of 252 square feet is about 15.87 ft (15 ft 10 in) on each side, so edging all the way around takes 4 × √252 ≈ 63.5 ft.
- 252 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √252 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √252 as its space diagonal.
- Since √252 = 6√7, a length of √252 is exactly 6 copies of the length √7 laid end to end.
Square roots near √252 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √249 | √249 | 15.7797 | No |
| √250 | 5√10 | 15.8114 | No |
| √251 | √251 | 15.8430 | No |
| √252 | 6√7 | 15.8745 | No |
| √253 | √253 | 15.9060 | No |
| √254 | √254 | 15.9374 | No |
| √255 | √255 | 15.9687 | No |
- The cube root of 252 is about 6.316360.
- Because 252 = 4 × 63, the root is twice √63: 2 × 7.937254 ≈ 15.874508.
Frequently asked questions
What is the square root of 252?
The square root of 252 is 6√7 in simplest radical form, which is about 15.8745078664. The negative root, −15.874508, also squares to 252.
Is the square root of 252 rational or irrational?
Irrational. 252 is not a perfect square — it falls between 225 and 256 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √252 be simplified?
Yes. The largest perfect square dividing 252 is 36, so √252 = √36 × √7 = 6√7.
What is √252 rounded to two decimal places?
√252 ≈ 15.87 to two decimal places (15.9 to one, 15.875 to three). Check: 15.87² = 251.8569, close to 252.