√138 at a glance
- Exact value
- √138
- Decimal (10 places)
- 11.7473401245
- Rounded
- 11.7 · 11.75 · 11.747
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.747340
- Prime factorization
- 2 × 3 × 23
- Cube root
- 5.167649
How to simplify √138
The prime factorization of 138 is 2 × 3 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √138 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 138, 2, 3 and 23 appear an odd number of times, so √138 is irrational and 11.7473401245 is a rounded value.
Where √138 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √138 lies between 11 and 12. 138 is 17 above 121 and 6 below 144, so the root is closer to 12.
- Straight line between 121 and 144: 11.7391 (0.07% low)
- Tangent from 11, i.e. 11 + 17 ÷ 22: 11.7727 (0.22% high)
- Tangent from 12, i.e. 12 − 6 ÷ 24: 11.7500 (0.02% high)
For √138 the tangent at 12 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 138 is just 6 below 144.
Finding √138 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, 12 (12² = 144):
| Step | Guess x | 138 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 12.0000000000 | 11.5000000000 | 11.7500000000 | 2 |
| 2 | 11.7500000000 | 11.7446808511 | 11.7473404255 | 6 |
| 3 | 11.7473404255 | 11.7473398234 | 11.7473401245 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √138 = 11.7473401245 to every decimal shown.
√138 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √138 the pattern is [11; 1, 2, 1, 22] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √138 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 | 7.5 × 10⁻¹ |
| 12/1 | 12.0000000000 | 2.5 × 10⁻¹ |
| 35/3 | 11.6666666667 | 8.1 × 10⁻² |
| 47/4 | 11.7500000000 | 2.7 × 10⁻³ |
| 1,069/91 | 11.7472527473 | 8.7 × 10⁻⁵ |
| 1,116/95 | 11.7473684211 | 2.8 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 138y² = 1. Its smallest solution in positive whole numbers is x = 47, y = 4.
√138 in geometry and everyday measurements
- A square room or garden bed covering 138 square feet measures about 11.75 ft (11 ft 9 in) along each wall.
- 138 is not a sum of two whole-number squares — the prime factor 3 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √138 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 11 box, because 1² + 4² + 11² = 138.
Square roots near √138 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √135 | 3√15 | 11.6190 | No |
| √136 | 2√34 | 11.6619 | No |
| √137 | √137 | 11.7047 | No |
| √138 | √138 | 11.7473 | No |
| √139 | √139 | 11.7898 | No |
| √140 | 2√35 | 11.8322 | No |
| √141 | √141 | 11.8743 | No |
- The cube root of 138 is about 5.167649.
- Four times the radicand doubles the root: √552 = 2 × √138 ≈ 23.49468.
Frequently asked questions
What is the square root of 138?
The square root of 138 is √138, about 11.7473401245. The negative root, −11.747340, also squares to 138.
Is the square root of 138 rational or irrational?
Irrational. 138 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 √138 be simplified?
No. 138 = 2 × 3 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √138 rounded to two decimal places?
√138 ≈ 11.75 to two decimal places (11.7 to one, 11.747 to three). Check: 11.75² = 138.0625, close to 138.