√950 at a glance
- Exact value
- 5√38
- Decimal (10 places)
- 30.8220700148
- Rounded
- 30.8 · 30.82 · 30.822
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.822070
- Prime factorization
- 2 × 5² × 19
- Cube root
- 9.830476
How to simplify √950
Look for the largest perfect square that divides 950. Here it is 25 (5²), because 950 = 25 × 38 and 38 has no square factor left:
The prime factorization tells the same story: 950 = 2 × 5² × 19. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 × 19 stays inside.
Check: (5√38)² = 5² × 38 = 25 × 38 = 950. As a decimal, 5√38 = 5 × 6.164414003 ≈ 30.8220700148.
Where √950 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √950 lies between 30 and 31. 950 is 50 above 900 and 11 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.8197 (0.01% low)
- Tangent from 30, i.e. 30 + 50 ÷ 60: 30.8333 (0.04% high)
- Tangent from 31, i.e. 31 − 11 ÷ 62: 30.8226 (0% high)
For √950 the tangent at 31 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 950 is just 11 below 961.
Finding √950 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, 31 (31² = 961):
| Step | Guess x | 950 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.6451612903 | 30.8225806452 | 3 |
| 2 | 30.8225806452 | 30.8215593930 | 30.8220700191 | 8 |
| 3 | 30.8220700191 | 30.8220700106 | 30.8220700148 | 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 √950 = 30.8220700148 to every decimal shown.
√950 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √950 the pattern is [30; 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √950 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 8.2 × 10⁻¹ |
| 31/1 | 31.0000000000 | 1.8 × 10⁻¹ |
| 154/5 | 30.8000000000 | 2.2 × 10⁻² |
| 185/6 | 30.8333333333 | 1.1 × 10⁻² |
| 339/11 | 30.8181818182 | 3.9 × 10⁻³ |
| 524/17 | 30.8235294118 | 1.5 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 950y² = 1. Its smallest solution in positive whole numbers is x = 202,501, y = 6,570.
√950 in geometry and everyday measurements
- 950 square feet is 88.3 m². Laid out as a square — a small house footprint or a lot — it is about 30.82 ft (30 ft 10 in) on a side.
- 950 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √950 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 30 box, because 1² + 7² + 30² = 950.
- Since √950 = 5√38, a length of √950 is exactly 5 copies of the length √38 laid end to end.
Square roots near √950 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √947 | √947 | 30.7734 | No |
| √948 | 2√237 | 30.7896 | No |
| √949 | √949 | 30.8058 | No |
| √950 | 5√38 | 30.8221 | No |
| √951 | √951 | 30.8383 | No |
| √952 | 2√238 | 30.8545 | No |
| √953 | √953 | 30.8707 | No |
- The cube root of 950 is about 9.830476.
- Squaring undoes the root: (√950)² = 950, while 950² = 902,500 — the number whose square root is 950.
Frequently asked questions
What is the square root of 950?
The square root of 950 is 5√38 in simplest radical form, which is about 30.8220700148. The negative root, −30.822070, also squares to 950.
Is the square root of 950 rational or irrational?
Irrational. 950 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √950 be simplified?
Yes. The largest perfect square dividing 950 is 25, so √950 = √25 × √38 = 5√38.
What is √950 rounded to two decimal places?
√950 ≈ 30.82 to two decimal places (30.8 to one, 30.822 to three). Check: 30.82² = 949.8724, close to 950.