√35 at a glance
- Exact value
- √35
- Decimal (10 places)
- 5.9160797831
- Rounded
- 5.9 · 5.92 · 5.916
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.916080
- Prime factorization
- 5 × 7
- Cube root
- 3.271066
How to simplify √35
The prime factorization of 35 is 5 × 7. Every prime appears only once, so there is no pair to bring outside the radical — √35 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 35, 5 and 7 appear an odd number of times, so √35 is irrational and 5.9160797831 is a rounded value.
Where √35 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √35 lies between 5 and 6. 35 is 10 above 25 and 1 below 36, so the root is closer to 6.
- Straight line between 25 and 36: 5.9091 (0.12% low)
- Tangent from 5, i.e. 5 + 10 ÷ 10: 6.0000 (1.42% high)
- Tangent from 6, i.e. 6 − 1 ÷ 12: 5.9167 (0.01% high)
For √35 the tangent at 6 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 35 is just 1 below 36.
Finding √35 with the Babylonian method
If a guess is too big, 35 ÷ 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√35) in one step.
Start from the nearest whole number, 6 (6² = 36):
| Step | Guess x | 35 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 6.0000000000 | 5.8333333333 | 5.9166666667 | 3 |
| 2 | 5.9166666667 | 5.9154929577 | 5.9160798122 | 7 |
| 3 | 5.9160798122 | 5.9160797540 | 5.9160797831 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √35 = 5.9160797831 to every decimal shown.
√35 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √35 the pattern is [5; 1, 10] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √35 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 |
|---|---|---|
| 5/1 | 5.0000000000 | 9.2 × 10⁻¹ |
| 6/1 | 6.0000000000 | 8.4 × 10⁻² |
| 65/11 | 5.9090909091 | 7.0 × 10⁻³ |
| 71/12 | 5.9166666667 | 5.9 × 10⁻⁴ |
| 775/131 | 5.9160305344 | 4.9 × 10⁻⁵ |
| 846/143 | 5.9160839161 | 4.1 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 35y² = 1. Its smallest solution in positive whole numbers is x = 6, y = 1.
√35 in geometry and everyday measurements
- A square room or garden bed covering 35 square feet measures about 5.92 ft (5 ft 11 in) along each wall.
- 35 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √35 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 5 box, because 1² + 3² + 5² = 35.
Square roots near √35 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √32 | 4√2 | 5.6569 | No |
| √33 | √33 | 5.7446 | No |
| √34 | √34 | 5.8310 | No |
| √35 | √35 | 5.9161 | No |
| √36 | 6 | 6.0000 | Yes |
| √37 | √37 | 6.0828 | No |
| √38 | √38 | 6.1644 | No |
- The cube root of 35 is about 3.271066.
- Four times the radicand doubles the root: √140 = 2 × √35 ≈ 11.83216.
Frequently asked questions
What is the square root of 35?
The square root of 35 is √35, about 5.9160797831. The negative root, −5.916080, also squares to 35.
Is the square root of 35 rational or irrational?
Irrational. 35 is not a perfect square — it falls between 25 and 36 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √35 be simplified?
No. 35 = 5 × 7 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √35 rounded to two decimal places?
√35 ≈ 5.92 to two decimal places (5.9 to one, 5.916 to three). Check: 5.92² = 35.0464, close to 35.