√50 at a glance
- Exact value
- 5√2
- Decimal (10 places)
- 7.0710678119
- Rounded
- 7.1 · 7.07 · 7.071
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.071068
- Prime factorization
- 2 × 5²
- Cube root
- 3.684031
How to simplify √50
Look for the largest perfect square that divides 50. Here it is 25 (5²), because 50 = 25 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 50 = 2 × 5². Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 stays inside.
Check: (5√2)² = 5² × 2 = 25 × 2 = 50. As a decimal, 5√2 = 5 × 1.4142135624 ≈ 7.0710678119.
Where √50 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √50 lies between 7 and 8. 50 is 1 above 49 and 14 below 64, so the root is closer to 7.
- Straight line between 49 and 64: 7.0667 (0.06% low)
- Tangent from 7, i.e. 7 + 1 ÷ 14: 7.0714 (0.01% high)
- Tangent from 8, i.e. 8 − 14 ÷ 16: 7.1250 (0.76% high)
For √50 the tangent at 7 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 50 is just 1 above 49.
Finding √50 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, 7 (7² = 49):
| Step | Guess x | 50 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 7.0000000000 | 7.1428571429 | 7.0714285714 | 3 |
| 2 | 7.0714285714 | 7.0707070707 | 7.0710678211 | 8 |
| 3 | 7.0710678211 | 7.0710678027 | 7.0710678119 | 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 √50 = 7.0710678119 to every decimal shown.
√50 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √50 the pattern is [7; 14] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 50 is one more than a perfect square (7² + 1). A pattern that never ends is one more proof that √50 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 | 7.1 × 10⁻² |
| 99/14 | 7.0714285714 | 3.6 × 10⁻⁴ |
| 1,393/197 | 7.0710659898 | 1.8 × 10⁻⁶ |
| 19,601/2,772 | 7.0710678211 | 9.2 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 50y² = 1. Its smallest solution in positive whole numbers is x = 99, y = 14. Because the period is odd, the equation with −1 on the right also has a solution: 7² − 50 × 1² = −1.
√50 in geometry and everyday measurements
- A square room or garden bed covering 50 square feet measures about 7.07 ft (7 ft 1 in) along each wall.
- 50 = 1² + 7² = 5² + 5², so by the Pythagorean theorem √50 is the diagonal of rectangles measuring 1 × 7 and 5 × 5 — and the distance between the points (0, 0) and (1, 7) on a grid.
- Since √50 = 5√2, a length of √50 is exactly 5 copies of the length √2 laid end to end.
Square roots near √50 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √47 | √47 | 6.8557 | No |
| √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 |
- The cube root of 50 is about 3.684031.
- Four times the radicand doubles the root: √200 = 2 × √50 ≈ 14.142136.
Frequently asked questions
What is the square root of 50?
The square root of 50 is 5√2 in simplest radical form, which is about 7.0710678119. The negative root, −7.071068, also squares to 50.
Is the square root of 50 rational or irrational?
Irrational. 50 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 √50 be simplified?
Yes. The largest perfect square dividing 50 is 25, so √50 = √25 × √2 = 5√2.
What is √50 rounded to two decimal places?
√50 ≈ 7.07 to two decimal places (7.1 to one, 7.071 to three). Check: 7.07² = 49.9849, close to 50.