√62 at a glance
- Exact value
- √62
- Decimal (10 places)
- 7.8740078740
- Rounded
- 7.9 · 7.87 · 7.874
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.874008
- Prime factorization
- 2 × 31
- Cube root
- 3.957892
How to simplify √62
The prime factorization of 62 is 2 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √62 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 62, 2 and 31 appear an odd number of times, so √62 is irrational and 7.8740078740 is a rounded value.
Where √62 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √62 lies between 7 and 8. 62 is 13 above 49 and 2 below 64, so the root is closer to 8.
- Straight line between 49 and 64: 7.8667 (0.09% low)
- Tangent from 7, i.e. 7 + 13 ÷ 14: 7.9286 (0.69% high)
- Tangent from 8, i.e. 8 − 2 ÷ 16: 7.8750 (0.01% high)
For √62 the tangent at 8 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 62 is just 2 below 64.
Finding √62 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 8 (8² = 64):
| Step | Guess x | 62 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 8.0000000000 | 7.7500000000 | 7.8750000000 | 3 |
| 2 | 7.8750000000 | 7.8730158730 | 7.8740079365 | 7 |
| 3 | 7.8740079365 | 7.8740078115 | 7.8740078740 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √62 = 7.8740078740 to every decimal shown.
√62 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √62 the pattern is [7; 1, 6, 1, 14] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √62 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 |
|---|---|---|
| 7/1 | 7.0000000000 | 8.7 × 10⁻¹ |
| 8/1 | 8.0000000000 | 1.3 × 10⁻¹ |
| 55/7 | 7.8571428571 | 1.7 × 10⁻² |
| 63/8 | 7.8750000000 | 9.9 × 10⁻⁴ |
| 937/119 | 7.8739495798 | 5.8 × 10⁻⁵ |
| 1,000/127 | 7.8740157480 | 7.9 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 62y² = 1. Its smallest solution in positive whole numbers is x = 63, y = 8.
√62 in geometry and everyday measurements
- A square room or garden bed covering 62 square feet measures about 7.87 ft (7 ft 10 in) along each wall.
- 62 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √62 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 6 box, because 1² + 5² + 6² = 62.
Square roots near √62 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √59 | √59 | 7.6811 | No |
| √60 | 2√15 | 7.7460 | No |
| √61 | √61 | 7.8102 | No |
| √62 | √62 | 7.8740 | No |
| √63 | 3√7 | 7.9373 | No |
| √64 | 8 | 8.0000 | Yes |
| √65 | √65 | 8.0623 | No |
- The cube root of 62 is about 3.957892.
- Four times the radicand doubles the root: √248 = 2 × √62 ≈ 15.748016.
Frequently asked questions
What is the square root of 62?
The square root of 62 is √62, about 7.8740078740. The negative root, −7.874008, also squares to 62.
Is the square root of 62 rational or irrational?
Irrational. 62 is not a perfect square — it falls between 49 and 64 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √62 be simplified?
No. 62 = 2 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √62 rounded to two decimal places?
√62 ≈ 7.87 to two decimal places (7.9 to one, 7.874 to three). Check: 7.87² = 61.9369, close to 62.