√537 at a glance
- Exact value
- √537
- Decimal (10 places)
- 23.1732604525
- Rounded
- 23.2 · 23.17 · 23.173
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.173260
- Prime factorization
- 3 × 179
- Cube root
- 8.128145
How to simplify √537
The prime factorization of 537 is 3 × 179. Every prime appears only once, so there is no pair to bring outside the radical — √537 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 537, 3 and 179 appear an odd number of times, so √537 is irrational and 23.1732604525 is a rounded value.
Where √537 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √537 lies between 23 and 24. 537 is 8 above 529 and 39 below 576, so the root is closer to 23.
- Straight line between 529 and 576: 23.1702 (0.01% low)
- Tangent from 23, i.e. 23 + 8 ÷ 46: 23.1739 (0% high)
- Tangent from 24, i.e. 24 − 39 ÷ 48: 23.1875 (0.06% high)
For √537 the tangent at 23 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 537 is just 8 above 529.
Finding √537 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 537: following the tangent line down to zero simplifies to averaging x with 537 ÷ x.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 537 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 23.3478260870 | 23.1739130435 | 3 |
| 2 | 23.1739130435 | 23.1726078799 | 23.1732604617 | 8 |
| 3 | 23.1732604617 | 23.1732604433 | 23.1732604525 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √537 = 23.1732604525 to every decimal shown.
√537 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √537 the pattern is [23; 5, 1, 3, 2, 1, 1, 1, 2, 1, 14, 1, 2, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √537 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 |
|---|---|---|
| 23/1 | 23.0000000000 | 1.7 × 10⁻¹ |
| 116/5 | 23.2000000000 | 2.7 × 10⁻² |
| 139/6 | 23.1666666667 | 6.6 × 10⁻³ |
| 533/23 | 23.1739130435 | 6.5 × 10⁻⁴ |
| 1,205/52 | 23.1730769231 | 1.8 × 10⁻⁴ |
| 1,738/75 | 23.1733333333 | 7.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 537y² = 1. Its smallest solution in positive whole numbers is x = 192,349,463, y = 8,300,492.
√537 in geometry and everyday measurements
- A square garage floor of 537 square feet measures about 23.17 ft (23 ft 2 in) per side, and its corner-to-corner diagonal is √1074 ≈ 32.8 ft.
- 537 is not a sum of two whole-number squares — the prime factor 3 and 179 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √537 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 23 box, because 2² + 2² + 23² = 537.
Square roots near √537 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √534 | √534 | 23.1084 | No |
| √535 | √535 | 23.1301 | No |
| √536 | 2√134 | 23.1517 | No |
| √537 | √537 | 23.1733 | No |
| √538 | √538 | 23.1948 | No |
| √539 | 7√11 | 23.2164 | No |
| √540 | 6√15 | 23.2379 | No |
- The cube root of 537 is about 8.128145.
- Squaring undoes the root: (√537)² = 537, while 537² = 288,369 — the number whose square root is 537.
Frequently asked questions
What is the square root of 537?
The square root of 537 is √537, about 23.1732604525. The negative root, −23.173260, also squares to 537.
Is the square root of 537 rational or irrational?
Irrational. 537 is not a perfect square — it falls between 529 and 576 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √537 be simplified?
No. 537 = 3 × 179 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √537 rounded to two decimal places?
√537 ≈ 23.17 to two decimal places (23.2 to one, 23.173 to three). Check: 23.17² = 536.8489, close to 537.