√15 at a glance
- Exact value
- √15
- Decimal (10 places)
- 3.8729833462
- Rounded
- 3.9 · 3.87 · 3.873
- Perfect square?
- No — between 3² and 4²
- Rational?
- Irrational
- Both square roots
- ±3.872983
- Prime factorization
- 3 × 5
- Cube root
- 2.466212
How to simplify √15
The prime factorization of 15 is 3 × 5. Every prime appears only once, so there is no pair to bring outside the radical — √15 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 15, 3 and 5 appear an odd number of times, so √15 is irrational and 3.8729833462 is a rounded value.
Where √15 sits between perfect squares
9 = 3² and 16 = 4² are the nearest perfect squares, so √15 lies between 3 and 4. 15 is 6 above 9 and 1 below 16, so the root is closer to 4.
- Straight line between 9 and 16: 3.8571 (0.41% low)
- Tangent from 3, i.e. 3 + 6 ÷ 6: 4.0000 (3.28% high)
- Tangent from 4, i.e. 4 − 1 ÷ 8: 3.8750 (0.05% high)
For √15 the tangent at 4 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 15 is just 1 below 16.
Finding √15 with the Babylonian method
If a guess is too big, 15 ÷ 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√15) in one step.
Start from the nearest whole number, 4 (4² = 16):
| Step | Guess x | 15 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 4.0000000000 | 3.7500000000 | 3.8750000000 | 2 |
| 2 | 3.8750000000 | 3.8709677419 | 3.8729838710 | 6 |
| 3 | 3.8729838710 | 3.8729828214 | 3.8729833462 | 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 √15 = 3.8729833462 to every decimal shown.
√15 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √15 the pattern is [3; 1, 6] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √15 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 |
|---|---|---|
| 3/1 | 3.0000000000 | 8.7 × 10⁻¹ |
| 4/1 | 4.0000000000 | 1.3 × 10⁻¹ |
| 27/7 | 3.8571428571 | 1.6 × 10⁻² |
| 31/8 | 3.8750000000 | 2.0 × 10⁻³ |
| 213/55 | 3.8727272727 | 2.6 × 10⁻⁴ |
| 244/63 | 3.8730158730 | 3.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 15y² = 1. Its smallest solution in positive whole numbers is x = 4, y = 1.
√15 in geometry and everyday measurements
- A square tile with an area of 15 square inches has sides about 3.873 in long.
- 15 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 √15 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 √15 as its space diagonal.
Square roots near √15 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √12 | 2√3 | 3.4641 | No |
| √13 | √13 | 3.6056 | No |
| √14 | √14 | 3.7417 | No |
| √15 | √15 | 3.8730 | No |
| √16 | 4 | 4.0000 | Yes |
| √17 | √17 | 4.1231 | No |
| √18 | 3√2 | 4.2426 | No |
- The cube root of 15 is about 2.466212.
- Four times the radicand doubles the root: √60 = 2 × √15 ≈ 7.745967.
Frequently asked questions
What is the square root of 15?
The square root of 15 is √15, about 3.8729833462. The negative root, −3.872983, also squares to 15.
Is the square root of 15 rational or irrational?
Irrational. 15 is not a perfect square — it falls between 9 and 16 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √15 be simplified?
No. 15 = 3 × 5 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √15 rounded to two decimal places?
√15 ≈ 3.87 to two decimal places (3.9 to one, 3.873 to three). Check: 3.87² = 14.9769, close to 15.