√58 at a glance
- Exact value
- √58
- Decimal (10 places)
- 7.6157731059
- Rounded
- 7.6 · 7.62 · 7.616
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.615773
- Prime factorization
- 2 × 29
- Cube root
- 3.870877
How to simplify √58
The prime factorization of 58 is 2 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √58 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 58, 2 and 29 appear an odd number of times, so √58 is irrational and 7.6157731059 is a rounded value.
Where √58 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √58 lies between 7 and 8. 58 is 9 above 49 and 6 below 64, so the root is closer to 8.
- Straight line between 49 and 64: 7.6000 (0.21% low)
- Tangent from 7, i.e. 7 + 9 ÷ 14: 7.6429 (0.36% high)
- Tangent from 8, i.e. 8 − 6 ÷ 16: 7.6250 (0.12% high)
For √58 the tangent at 8 wins, missing by only 0.0092. Tangent estimates shine when the number sits close to a perfect square — here 58 is just 6 below 64.
Finding √58 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, 8 (8² = 64):
| Step | Guess x | 58 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 8.0000000000 | 7.2500000000 | 7.6250000000 | 2 |
| 2 | 7.6250000000 | 7.6065573770 | 7.6157786885 | 5 |
| 3 | 7.6157786885 | 7.6157675232 | 7.6157731059 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √58 = 7.6157731059 to every decimal shown.
√58 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √58 the pattern is [7; 1, 1, 1, 1, 1, 1, 14] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √58 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 |
|---|---|---|
| 7/1 | 7.0000000000 | 6.2 × 10⁻¹ |
| 8/1 | 8.0000000000 | 3.8 × 10⁻¹ |
| 15/2 | 7.5000000000 | 1.2 × 10⁻¹ |
| 23/3 | 7.6666666667 | 5.1 × 10⁻² |
| 38/5 | 7.6000000000 | 1.6 × 10⁻² |
| 61/8 | 7.6250000000 | 9.2 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 58y² = 1. Its smallest solution in positive whole numbers is x = 19,603, y = 2,574. Because the period is odd, the equation with −1 on the right also has a solution: 99² − 58 × 13² = −1.
√58 in geometry and everyday measurements
- A square room or garden bed covering 58 square feet measures about 7.62 ft (7 ft 7 in) along each wall.
- 58 = 3² + 7², so by the Pythagorean theorem √58 is the diagonal of a 3 × 7 rectangle — and the distance between the points (0, 0) and (3, 7) on a grid.
Square roots near √58 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √55 | √55 | 7.4162 | No |
| √56 | 2√14 | 7.4833 | No |
| √57 | √57 | 7.5498 | No |
| √58 | √58 | 7.6158 | No |
| √59 | √59 | 7.6811 | No |
| √60 | 2√15 | 7.7460 | No |
| √61 | √61 | 7.8102 | No |
- The cube root of 58 is about 3.870877.
- Four times the radicand doubles the root: √232 = 2 × √58 ≈ 15.231546.
Frequently asked questions
What is the square root of 58?
The square root of 58 is √58, about 7.6157731059. The negative root, −7.615773, also squares to 58.
Is the square root of 58 rational or irrational?
Irrational. 58 is not a perfect square — it falls between 49 and 64 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √58 be simplified?
No. 58 = 2 × 29 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √58 rounded to two decimal places?
√58 ≈ 7.62 to two decimal places (7.6 to one, 7.616 to three). Check: 7.62² = 58.0644, close to 58.