√158 at a glance
- Exact value
- √158
- Decimal (10 places)
- 12.5698050900
- Rounded
- 12.6 · 12.57 · 12.570
- Perfect square?
- No — between 12² and 13²
- Rational?
- Irrational
- Both square roots
- ±12.569805
- Prime factorization
- 2 × 79
- Cube root
- 5.406120
How to simplify √158
The prime factorization of 158 is 2 × 79. Every prime appears only once, so there is no pair to bring outside the radical — √158 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 158, 2 and 79 appear an odd number of times, so √158 is irrational and 12.5698050900 is a rounded value.
Where √158 sits between perfect squares
144 = 12² and 169 = 13² are the nearest perfect squares, so √158 lies between 12 and 13. 158 is 14 above 144 and 11 below 169, so the root is closer to 13.
- Straight line between 144 and 169: 12.5600 (0.08% low)
- Tangent from 12, i.e. 12 + 14 ÷ 24: 12.5833 (0.11% high)
- Tangent from 13, i.e. 13 − 11 ÷ 26: 12.5769 (0.06% high)
For √158 the tangent at 13 wins, missing by only 0.0071. Tangent estimates shine when the number sits close to a perfect square — here 158 is just 11 below 169.
Finding √158 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, 13 (13² = 169):
| Step | Guess x | 158 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 12.1538461538 | 12.5769230769 | 2 |
| 2 | 12.5769230769 | 12.5626911315 | 12.5698071042 | 5 |
| 3 | 12.5698071042 | 12.5698030757 | 12.5698050900 | 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 √158 = 12.5698050900 to every decimal shown.
√158 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √158 the pattern is [12; 1, 1, 3, 12, 3, 1, 1, 24] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √158 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 | 5.7 × 10⁻¹ |
| 13/1 | 13.0000000000 | 4.3 × 10⁻¹ |
| 25/2 | 12.5000000000 | 7.0 × 10⁻² |
| 88/7 | 12.5714285714 | 1.6 × 10⁻³ |
| 1,081/86 | 12.5697674419 | 3.8 × 10⁻⁵ |
| 3,331/265 | 12.5698113208 | 6.2 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 158y² = 1. Its smallest solution in positive whole numbers is x = 7,743, y = 616.
√158 in geometry and everyday measurements
- A square room or garden bed covering 158 square feet measures about 12.57 ft (12 ft 7 in) along each wall.
- 158 is not a sum of two whole-number squares — the prime factor 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √158 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 6 × 11 box, because 1² + 6² + 11² = 158.
Square roots near √158 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √155 | √155 | 12.4499 | No |
| √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 |
- The cube root of 158 is about 5.406120.
- Four times the radicand doubles the root: √632 = 2 × √158 ≈ 25.13961.
Frequently asked questions
What is the square root of 158?
The square root of 158 is √158, about 12.5698050900. The negative root, −12.569805, also squares to 158.
Is the square root of 158 rational or irrational?
Irrational. 158 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 √158 be simplified?
No. 158 = 2 × 79 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √158 rounded to two decimal places?
√158 ≈ 12.57 to two decimal places (12.6 to one, 12.570 to three). Check: 12.57² = 158.0049, close to 158.