√159 at a glance
- Exact value
- √159
- Decimal (10 places)
- 12.6095202129
- Rounded
- 12.6 · 12.61 · 12.610
- Perfect square?
- No — between 12² and 13²
- Rational?
- Irrational
- Both square roots
- ±12.609520
- Prime factorization
- 3 × 53
- Cube root
- 5.417502
How to simplify √159
The prime factorization of 159 is 3 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √159 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 159, 3 and 53 appear an odd number of times, so √159 is irrational and 12.6095202129 is a rounded value.
Where √159 sits between perfect squares
144 = 12² and 169 = 13² are the nearest perfect squares, so √159 lies between 12 and 13. 159 is 15 above 144 and 10 below 169, so the root is closer to 13.
- Straight line between 144 and 169: 12.6000 (0.08% low)
- Tangent from 12, i.e. 12 + 15 ÷ 24: 12.6250 (0.12% high)
- Tangent from 13, i.e. 13 − 10 ÷ 26: 12.6154 (0.05% high)
For √159 the tangent at 13 wins, missing by only 0.0059. Tangent estimates shine when the number sits close to a perfect square — here 159 is just 10 below 169.
Finding √159 with the Babylonian method
If a guess is too big, 159 ÷ 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√159) in one step.
Start from the nearest whole number, 13 (13² = 169):
| Step | Guess x | 159 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 12.2307692308 | 12.6153846154 | 2 |
| 2 | 12.6153846154 | 12.6036585366 | 12.6095215760 | 5 |
| 3 | 12.6095215760 | 12.6095188499 | 12.6095202129 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √159 = 12.6095202129 to every decimal shown.
√159 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √159 the pattern is [12; 1, 1, 1, 1, 3, 1, 1, 1, 1, 24] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √159 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 |
|---|---|---|
| 12/1 | 12.0000000000 | 6.1 × 10⁻¹ |
| 13/1 | 13.0000000000 | 3.9 × 10⁻¹ |
| 25/2 | 12.5000000000 | 1.1 × 10⁻¹ |
| 38/3 | 12.6666666667 | 5.7 × 10⁻² |
| 63/5 | 12.6000000000 | 9.5 × 10⁻³ |
| 227/18 | 12.6111111111 | 1.6 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 159y² = 1. Its smallest solution in positive whole numbers is x = 1,324, y = 105.
√159 in geometry and everyday measurements
- A square room or garden bed covering 159 square feet measures about 12.61 ft (12 ft 7 in) along each wall.
- 159 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 √159 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √159 as its space diagonal.
Square roots near √159 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √156 | 2√39 | 12.4900 | No |
| √157 | √157 | 12.5300 | No |
| √158 | √158 | 12.5698 | No |
| √159 | √159 | 12.6095 | No |
| √160 | 4√10 | 12.6491 | No |
| √161 | √161 | 12.6886 | No |
| √162 | 9√2 | 12.7279 | No |
- The cube root of 159 is about 5.417502.
- Four times the radicand doubles the root: √636 = 2 × √159 ≈ 25.21904.
Frequently asked questions
What is the square root of 159?
The square root of 159 is √159, about 12.6095202129. The negative root, −12.609520, also squares to 159.
Is the square root of 159 rational or irrational?
Irrational. 159 is not a perfect square — it falls between 144 and 169 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √159 be simplified?
No. 159 = 3 × 53 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √159 rounded to two decimal places?
√159 ≈ 12.61 to two decimal places (12.6 to one, 12.610 to three). Check: 12.61² = 159.0121, close to 159.