√51 at a glance
- Exact value
- √51
- Decimal (10 places)
- 7.1414284285
- Rounded
- 7.1 · 7.14 · 7.141
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.141428
- Prime factorization
- 3 × 17
- Cube root
- 3.708430
How to simplify √51
The prime factorization of 51 is 3 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √51 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 51, 3 and 17 appear an odd number of times, so √51 is irrational and 7.1414284285 is a rounded value.
Where √51 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √51 lies between 7 and 8. 51 is 2 above 49 and 13 below 64, so the root is closer to 7.
- Straight line between 49 and 64: 7.1333 (0.11% low)
- Tangent from 7, i.e. 7 + 2 ÷ 14: 7.1429 (0.02% high)
- Tangent from 8, i.e. 8 − 13 ÷ 16: 7.1875 (0.65% high)
For √51 the tangent at 7 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 51 is just 2 above 49.
Finding √51 with the Babylonian method
If a guess is too big, 51 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√51) in one step.
Start from the nearest whole number, 7 (7² = 49):
| Step | Guess x | 51 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 7.0000000000 | 7.2857142857 | 7.1428571429 | 2 |
| 2 | 7.1428571429 | 7.1400000000 | 7.1414285714 | 6 |
| 3 | 7.1414285714 | 7.1414282857 | 7.1414284285 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √51 = 7.1414284285 to every decimal shown.
√51 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √51 the pattern is [7; 7, 14] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √51 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 | 1.4 × 10⁻¹ |
| 50/7 | 7.1428571429 | 1.4 × 10⁻³ |
| 707/99 | 7.1414141414 | 1.4 × 10⁻⁵ |
| 4,999/700 | 7.1414285714 | 1.4 × 10⁻⁷ |
| 70,693/9,899 | 7.1414284271 | 1.4 × 10⁻⁹ |
| 499,850/69,993 | 7.1414284286 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 51y² = 1. Its smallest solution in positive whole numbers is x = 50, y = 7.
√51 in geometry and everyday measurements
- A square room or garden bed covering 51 square feet measures about 7.14 ft (7 ft 2 in) along each wall.
- 51 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √51 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 7 box, because 1² + 1² + 7² = 51.
Square roots near √51 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √48 | 4√3 | 6.9282 | No |
| √49 | 7 | 7.0000 | Yes |
| √50 | 5√2 | 7.0711 | No |
| √51 | √51 | 7.1414 | No |
| √52 | 2√13 | 7.2111 | No |
| √53 | √53 | 7.2801 | No |
| √54 | 3√6 | 7.3485 | No |
- The cube root of 51 is about 3.708430.
- Four times the radicand doubles the root: √204 = 2 × √51 ≈ 14.282857.
Frequently asked questions
What is the square root of 51?
The square root of 51 is √51, about 7.1414284285. The negative root, −7.141428, also squares to 51.
Is the square root of 51 rational or irrational?
Irrational. 51 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 √51 be simplified?
No. 51 = 3 × 17 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √51 rounded to two decimal places?
√51 ≈ 7.14 to two decimal places (7.1 to one, 7.141 to three). Check: 7.14² = 50.9796, close to 51.