√141 at a glance
- Exact value
- √141
- Decimal (10 places)
- 11.8743420870
- Rounded
- 11.9 · 11.87 · 11.874
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.874342
- Prime factorization
- 3 × 47
- Cube root
- 5.204828
How to simplify √141
The prime factorization of 141 is 3 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √141 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 141, 3 and 47 appear an odd number of times, so √141 is irrational and 11.8743420870 is a rounded value.
Where √141 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √141 lies between 11 and 12. 141 is 20 above 121 and 3 below 144, so the root is closer to 12.
- Straight line between 121 and 144: 11.8696 (0.04% low)
- Tangent from 11, i.e. 11 + 20 ÷ 22: 11.9091 (0.29% high)
- Tangent from 12, i.e. 12 − 3 ÷ 24: 11.8750 (0.01% high)
For √141 the tangent at 12 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 141 is just 3 below 144.
Finding √141 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 141: following the tangent line down to zero simplifies to averaging x with 141 ÷ x.
Start from the nearest whole number, 12 (12² = 144):
| Step | Guess x | 141 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 12.0000000000 | 11.7500000000 | 11.8750000000 | 3 |
| 2 | 11.8750000000 | 11.8736842105 | 11.8743421053 | 7 |
| 3 | 11.8743421053 | 11.8743420688 | 11.8743420870 | 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 √141 = 11.8743420870 to every decimal shown.
√141 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √141 the pattern is [11; 1, 6, 1, 22] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √141 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 |
|---|---|---|
| 11/1 | 11.0000000000 | 8.7 × 10⁻¹ |
| 12/1 | 12.0000000000 | 1.3 × 10⁻¹ |
| 83/7 | 11.8571428571 | 1.7 × 10⁻² |
| 95/8 | 11.8750000000 | 6.6 × 10⁻⁴ |
| 2,173/183 | 11.8743169399 | 2.5 × 10⁻⁵ |
| 2,268/191 | 11.8743455497 | 3.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 141y² = 1. Its smallest solution in positive whole numbers is x = 95, y = 8.
√141 in geometry and everyday measurements
- A square room or garden bed covering 141 square feet measures about 11.87 ft (11 ft 10 in) along each wall.
- 141 is not a sum of two whole-number squares — the prime factor 3 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √141 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 11 box, because 2² + 4² + 11² = 141.
Square roots near √141 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √138 | √138 | 11.7473 | No |
| √139 | √139 | 11.7898 | No |
| √140 | 2√35 | 11.8322 | No |
| √141 | √141 | 11.8743 | No |
| √142 | √142 | 11.9164 | No |
| √143 | √143 | 11.9583 | No |
| √144 | 12 | 12.0000 | Yes |
- The cube root of 141 is about 5.204828.
- Four times the radicand doubles the root: √564 = 2 × √141 ≈ 23.748684.
Frequently asked questions
What is the square root of 141?
The square root of 141 is √141, about 11.8743420870. The negative root, −11.874342, also squares to 141.
Is the square root of 141 rational or irrational?
Irrational. 141 is not a perfect square — it falls between 121 and 144 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √141 be simplified?
No. 141 = 3 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √141 rounded to two decimal places?
√141 ≈ 11.87 to two decimal places (11.9 to one, 11.874 to three). Check: 11.87² = 140.8969, close to 141.