√38 at a glance
- Exact value
- √38
- Decimal (10 places)
- 6.1644140030
- Rounded
- 6.2 · 6.16 · 6.164
- Perfect square?
- No — between 6² and 7²
- Rational?
- Irrational
- Both square roots
- ±6.164414
- Prime factorization
- 2 × 19
- Cube root
- 3.361975
How to simplify √38
The prime factorization of 38 is 2 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √38 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 38, 2 and 19 appear an odd number of times, so √38 is irrational and 6.1644140030 is a rounded value.
Where √38 sits between perfect squares
36 = 6² and 49 = 7² are the nearest perfect squares, so √38 lies between 6 and 7. 38 is 2 above 36 and 11 below 49, so the root is closer to 6.
- Straight line between 36 and 49: 6.1538 (0.17% low)
- Tangent from 6, i.e. 6 + 2 ÷ 12: 6.1667 (0.04% high)
- Tangent from 7, i.e. 7 − 11 ÷ 14: 6.2143 (0.81% high)
For √38 the tangent at 6 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 38 is just 2 above 36.
Finding √38 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, 6 (6² = 36):
| Step | Guess x | 38 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 6.0000000000 | 6.3333333333 | 6.1666666667 | 2 |
| 2 | 6.1666666667 | 6.1621621622 | 6.1644144144 | 6 |
| 3 | 6.1644144144 | 6.1644135915 | 6.1644140030 | 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 √38 = 6.1644140030 to every decimal shown.
√38 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √38 the pattern is [6; 6, 12] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √38 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 |
|---|---|---|
| 6/1 | 6.0000000000 | 1.6 × 10⁻¹ |
| 37/6 | 6.1666666667 | 2.3 × 10⁻³ |
| 450/73 | 6.1643835616 | 3.0 × 10⁻⁵ |
| 2,737/444 | 6.1644144144 | 4.1 × 10⁻⁷ |
| 33,294/5,401 | 6.1644139974 | 5.6 × 10⁻⁹ |
| 202,501/32,850 | 6.1644140030 | 7.5 × 10⁻¹¹ |
The same fractions solve Pell’s equation, x² − 38y² = 1. Its smallest solution in positive whole numbers is x = 37, y = 6.
√38 in geometry and everyday measurements
- A square room or garden bed covering 38 square feet measures about 6.16 ft (6 ft 2 in) along each wall.
- 38 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √38 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 6 box, because 1² + 1² + 6² = 38.
Square roots near √38 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √35 | √35 | 5.9161 | No |
| √36 | 6 | 6.0000 | Yes |
| √37 | √37 | 6.0828 | No |
| √38 | √38 | 6.1644 | No |
| √39 | √39 | 6.2450 | No |
| √40 | 2√10 | 6.3246 | No |
| √41 | √41 | 6.4031 | No |
- The cube root of 38 is about 3.361975.
- Four times the radicand doubles the root: √152 = 2 × √38 ≈ 12.328828.
Frequently asked questions
What is the square root of 38?
The square root of 38 is √38, about 6.1644140030. The negative root, −6.164414, also squares to 38.
Is the square root of 38 rational or irrational?
Irrational. 38 is not a perfect square — it falls between 36 and 49 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √38 be simplified?
No. 38 = 2 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √38 rounded to two decimal places?
√38 ≈ 6.16 to two decimal places (6.2 to one, 6.164 to three). Check: 6.16² = 37.9456, close to 38.