√57 at a glance
- Exact value
- √57
- Decimal (10 places)
- 7.5498344353
- Rounded
- 7.5 · 7.55 · 7.550
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.549834
- Prime factorization
- 3 × 19
- Cube root
- 3.848501
How to simplify √57
The prime factorization of 57 is 3 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √57 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 57, 3 and 19 appear an odd number of times, so √57 is irrational and 7.5498344353 is a rounded value.
Where √57 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √57 lies between 7 and 8. 57 is 8 above 49 and 7 below 64, so the root is closer to 8.
- Straight line between 49 and 64: 7.5333 (0.22% low)
- Tangent from 7, i.e. 7 + 8 ÷ 14: 7.5714 (0.29% high)
- Tangent from 8, i.e. 8 − 7 ÷ 16: 7.5625 (0.17% high)
For √57 the tangent at 8 wins, missing by only 0.0127. Tangent estimates shine when the number sits close to a perfect square — here 57 is just 7 below 64.
Finding √57 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 57: following the tangent line down to zero simplifies to averaging x with 57 ÷ x.
Start from the nearest whole number, 8 (8² = 64):
| Step | Guess x | 57 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 8.0000000000 | 7.1250000000 | 7.5625000000 | 1 |
| 2 | 7.5625000000 | 7.5371900826 | 7.5498450413 | 4 |
| 3 | 7.5498450413 | 7.5498238292 | 7.5498344353 | 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 √57 = 7.5498344353 to every decimal shown.
√57 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √57 the pattern is [7; 1, 1, 4, 1, 1, 14] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √57 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 | 5.5 × 10⁻¹ |
| 8/1 | 8.0000000000 | 4.5 × 10⁻¹ |
| 15/2 | 7.5000000000 | 5.0 × 10⁻² |
| 68/9 | 7.5555555556 | 5.7 × 10⁻³ |
| 83/11 | 7.5454545455 | 4.4 × 10⁻³ |
| 151/20 | 7.5500000000 | 1.7 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 57y² = 1. Its smallest solution in positive whole numbers is x = 151, y = 20.
√57 in geometry and everyday measurements
- A square room or garden bed covering 57 square feet measures about 7.55 ft (7 ft 7 in) along each wall.
- 57 is not a sum of two whole-number squares — the prime factor 3 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √57 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 7 box, because 2² + 2² + 7² = 57.
Square roots near √57 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √54 | 3√6 | 7.3485 | No |
| √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 |
- The cube root of 57 is about 3.848501.
- Four times the radicand doubles the root: √228 = 2 × √57 ≈ 15.099669.
Frequently asked questions
What is the square root of 57?
The square root of 57 is √57, about 7.5498344353. The negative root, −7.549834, also squares to 57.
Is the square root of 57 rational or irrational?
Irrational. 57 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 √57 be simplified?
No. 57 = 3 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √57 rounded to two decimal places?
√57 ≈ 7.55 to two decimal places (7.5 to one, 7.550 to three). Check: 7.55² = 57.0025, close to 57.