√17 at a glance
- Exact value
- √17
- Decimal (10 places)
- 4.1231056256
- Rounded
- 4.1 · 4.12 · 4.123
- Perfect square?
- No — between 4² and 5²
- Rational?
- Irrational
- Both square roots
- ±4.123106
- Prime factorization
- 17
- Cube root
- 2.571282
How to simplify √17
17 is a prime number, so its only factors are 1 and 17. There is no perfect-square factor to pull out, which means √17 is already in its simplest radical form.
The square root of any prime is irrational. If √17 were a fraction a/b in lowest terms, then a² = 17b², so 17 would divide a — and then 17 would divide b too, contradicting “lowest terms.” That is why the decimal 4.1231056256 is only a rounded value.
Where √17 sits between perfect squares
16 = 4² and 25 = 5² are the nearest perfect squares, so √17 lies between 4 and 5. 17 is 1 above 16 and 8 below 25, so the root is closer to 4.
- Straight line between 16 and 25: 4.1111 (0.29% low)
- Tangent from 4, i.e. 4 + 1 ÷ 8: 4.1250 (0.05% high)
- Tangent from 5, i.e. 5 − 8 ÷ 10: 4.2000 (1.86% high)
For √17 the tangent at 4 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 17 is just 1 above 16.
Finding √17 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 17: following the tangent line down to zero simplifies to averaging x with 17 ÷ x.
Start from the nearest whole number, 4 (4² = 16):
| Step | Guess x | 17 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 4.0000000000 | 4.2500000000 | 4.1250000000 | 2 |
| 2 | 4.1250000000 | 4.1212121212 | 4.1231060606 | 6 |
| 3 | 4.1231060606 | 4.1231051906 | 4.1231056256 | 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 √17 = 4.1231056256 to every decimal shown.
√17 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √17 the pattern is [4; 8] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 17 is one more than a perfect square (4² + 1). A pattern that never ends is one more proof that √17 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 |
|---|---|---|
| 4/1 | 4.0000000000 | 1.2 × 10⁻¹ |
| 33/8 | 4.1250000000 | 1.9 × 10⁻³ |
| 268/65 | 4.1230769231 | 2.9 × 10⁻⁵ |
| 2,177/528 | 4.1231060606 | 4.3 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 17y² = 1. Its smallest solution in positive whole numbers is x = 33, y = 8. Because the period is odd, the equation with −1 on the right also has a solution: 4² − 17 × 1² = −1.
√17 in geometry and everyday measurements
- A square tile with an area of 17 square inches has sides about 4.123 in long.
- 17 = 1² + 4², so by the Pythagorean theorem √17 is the diagonal of a 1 × 4 rectangle — and the distance between the points (0, 0) and (1, 4) on a grid.
Square roots near √17 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √19 | √19 | 4.3589 | No |
| √20 | 2√5 | 4.4721 | No |
- The cube root of 17 is about 2.571282.
- Four times the radicand doubles the root: √68 = 2 × √17 ≈ 8.246211.
Frequently asked questions
What is the square root of 17?
The square root of 17 is √17, about 4.1231056256. The negative root, −4.123106, also squares to 17.
Is the square root of 17 rational or irrational?
Irrational. 17 is not a perfect square — it falls between 16 and 25 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √17 be simplified?
No. 17 is prime, so there is no perfect square to take out of the radical.
What is √17 rounded to two decimal places?
√17 ≈ 4.12 to two decimal places (4.1 to one, 4.123 to three). Check: 4.12² = 16.9744, close to 17.