√155 at a glance
- Exact value
- √155
- Decimal (10 places)
- 12.4498995980
- Rounded
- 12.4 · 12.45 · 12.450
- Perfect square?
- No — between 12² and 13²
- Rational?
- Irrational
- Both square roots
- ±12.449900
- Prime factorization
- 5 × 31
- Cube root
- 5.371685
How to simplify √155
The prime factorization of 155 is 5 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √155 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 155, 5 and 31 appear an odd number of times, so √155 is irrational and 12.4498995980 is a rounded value.
Where √155 sits between perfect squares
144 = 12² and 169 = 13² are the nearest perfect squares, so √155 lies between 12 and 13. 155 is 11 above 144 and 14 below 169, so the root is closer to 12.
- Straight line between 144 and 169: 12.4400 (0.08% low)
- Tangent from 12, i.e. 12 + 11 ÷ 24: 12.4583 (0.07% high)
- Tangent from 13, i.e. 13 − 14 ÷ 26: 12.4615 (0.09% high)
For √155 the tangent at 12 wins, missing by only 0.0084. Tangent estimates shine when the number sits close to a perfect square — here 155 is just 11 above 144.
Finding √155 with the Babylonian method
If a guess is too big, 155 ÷ 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√155) in one step.
Start from the nearest whole number, 12 (12² = 144):
| Step | Guess x | 155 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 12.0000000000 | 12.9166666667 | 12.4583333333 | 2 |
| 2 | 12.4583333333 | 12.4414715719 | 12.4499024526 | 5 |
| 3 | 12.4499024526 | 12.4498967434 | 12.4498995980 | 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 √155 = 12.4498995980 to every decimal shown.
√155 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √155 the pattern is [12; 2, 4, 2, 24] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √155 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 | 4.5 × 10⁻¹ |
| 25/2 | 12.5000000000 | 5.0 × 10⁻² |
| 112/9 | 12.4444444444 | 5.5 × 10⁻³ |
| 249/20 | 12.4500000000 | 1.0 × 10⁻⁴ |
| 6,088/489 | 12.4498977505 | 1.8 × 10⁻⁶ |
| 12,425/998 | 12.4498997996 | 2.0 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 155y² = 1. Its smallest solution in positive whole numbers is x = 249, y = 20.
√155 in geometry and everyday measurements
- A square room or garden bed covering 155 square feet measures about 12.45 ft (12 ft 5 in) along each wall.
- 155 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √155 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 11 box, because 3² + 5² + 11² = 155.
Square roots near √155 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √152 | 2√38 | 12.3288 | No |
| √153 | 3√17 | 12.3693 | No |
| √154 | √154 | 12.4097 | No |
| √155 | √155 | 12.4499 | No |
| √156 | 2√39 | 12.4900 | No |
| √157 | √157 | 12.5300 | No |
| √158 | √158 | 12.5698 | No |
- The cube root of 155 is about 5.371685.
- Four times the radicand doubles the root: √620 = 2 × √155 ≈ 24.899799.
Frequently asked questions
What is the square root of 155?
The square root of 155 is √155, about 12.4498995980. The negative root, −12.449900, also squares to 155.
Is the square root of 155 rational or irrational?
Irrational. 155 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 √155 be simplified?
No. 155 = 5 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √155 rounded to two decimal places?
√155 ≈ 12.45 to two decimal places (12.4 to one, 12.450 to three). Check: 12.45² = 155.0025, close to 155.