√29 at a glance
- Exact value
- √29
- Decimal (10 places)
- 5.3851648071
- Rounded
- 5.4 · 5.39 · 5.385
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.385165
- Prime factorization
- 29
- Cube root
- 3.072317
How to simplify √29
29 is a prime number, so its only factors are 1 and 29. There is no perfect-square factor to pull out, which means √29 is already in its simplest radical form.
The square root of any prime is irrational. If √29 were a fraction a/b in lowest terms, then a² = 29b², so 29 would divide a — and then 29 would divide b too, contradicting “lowest terms.” That is why the decimal 5.3851648071 is only a rounded value.
Where √29 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √29 lies between 5 and 6. 29 is 4 above 25 and 7 below 36, so the root is closer to 5.
- Straight line between 25 and 36: 5.3636 (0.4% low)
- Tangent from 5, i.e. 5 + 4 ÷ 10: 5.4000 (0.28% high)
- Tangent from 6, i.e. 6 − 7 ÷ 12: 5.4167 (0.58% high)
For √29 the tangent at 5 wins, missing by only 0.0148. Tangent estimates shine when the number sits close to a perfect square — here 29 is just 4 above 25.
Finding √29 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 29: following the tangent line down to zero simplifies to averaging x with 29 ÷ x.
Start from the nearest whole number, 5 (5² = 25):
| Step | Guess x | 29 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 5.0000000000 | 5.8000000000 | 5.4000000000 | 1 |
| 2 | 5.4000000000 | 5.3703703704 | 5.3851851852 | 4 |
| 3 | 5.3851851852 | 5.3851444292 | 5.3851648072 | all 10 shown |
The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √29 = 5.3851648071 to every decimal shown.
√29 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √29 the pattern is [5; 2, 1, 1, 2, 10] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √29 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 | 3.9 × 10⁻¹ |
| 11/2 | 5.5000000000 | 1.1 × 10⁻¹ |
| 16/3 | 5.3333333333 | 5.2 × 10⁻² |
| 27/5 | 5.4000000000 | 1.5 × 10⁻² |
| 70/13 | 5.3846153846 | 5.5 × 10⁻⁴ |
| 727/135 | 5.3851851852 | 2.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 29y² = 1. Its smallest solution in positive whole numbers is x = 9,801, y = 1,820. Because the period is odd, the equation with −1 on the right also has a solution: 70² − 29 × 13² = −1.
√29 in geometry and everyday measurements
- A square tile with an area of 29 square inches has sides about 5.385 in long.
- 29 = 2² + 5², so by the Pythagorean theorem √29 is the diagonal of a 2 × 5 rectangle — and the distance between the points (0, 0) and (2, 5) on a grid.
Square roots near √29 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √26 | √26 | 5.0990 | No |
| √27 | 3√3 | 5.1962 | No |
| √28 | 2√7 | 5.2915 | No |
| √29 | √29 | 5.3852 | No |
| √30 | √30 | 5.4772 | No |
| √31 | √31 | 5.5678 | No |
| √32 | 4√2 | 5.6569 | No |
- The cube root of 29 is about 3.072317.
- Four times the radicand doubles the root: √116 = 2 × √29 ≈ 10.77033.
Frequently asked questions
What is the square root of 29?
The square root of 29 is √29, about 5.3851648071. The negative root, −5.385165, also squares to 29.
Is the square root of 29 rational or irrational?
Irrational. 29 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 √29 be simplified?
No. 29 is prime, so there is no perfect square to take out of the radical.
What is √29 rounded to two decimal places?
√29 ≈ 5.39 to two decimal places (5.4 to one, 5.385 to three). Check: 5.39² = 29.0521, close to 29.