√507 at a glance
- Exact value
- 13√3
- Decimal (10 places)
- 22.5166604984
- Rounded
- 22.5 · 22.52 · 22.517
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.516660
- Prime factorization
- 3 × 13²
- Cube root
- 7.973873
How to simplify √507
Look for the largest perfect square that divides 507. Here it is 169 (13²), because 507 = 169 × 3 and 3 has no square factor left:
The prime factorization tells the same story: 507 = 3 × 13². Each pair of equal primes leaves the radical as one factor, so 13 comes out and 3 stays inside.
Check: (13√3)² = 13² × 3 = 169 × 3 = 507. As a decimal, 13√3 = 13 × 1.7320508076 ≈ 22.5166604984.
Where √507 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √507 lies between 22 and 23. 507 is 23 above 484 and 22 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.5111 (0.02% low)
- Tangent from 22, i.e. 22 + 23 ÷ 44: 22.5227 (0.03% high)
- Tangent from 23, i.e. 23 − 22 ÷ 46: 22.5217 (0.02% high)
For √507 the tangent at 23 wins, missing by only 0.0051. Tangent estimates shine when the number sits close to a perfect square — here 507 is just 22 below 529.
Finding √507 with the Babylonian method
If a guess is too big, 507 ÷ 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√507) in one step.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 507 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.0434782609 | 22.5217391304 | 2 |
| 2 | 22.5217391304 | 22.5115830116 | 22.5166610710 | 6 |
| 3 | 22.5166610710 | 22.5166599258 | 22.5166604984 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √507 = 22.5166604984 to every decimal shown.
√507 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √507 the pattern is [22; 1, 1, 14, 1, 1, 44] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √507 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 |
|---|---|---|
| 22/1 | 22.0000000000 | 5.2 × 10⁻¹ |
| 23/1 | 23.0000000000 | 4.8 × 10⁻¹ |
| 45/2 | 22.5000000000 | 1.7 × 10⁻² |
| 653/29 | 22.5172413793 | 5.8 × 10⁻⁴ |
| 698/31 | 22.5161290323 | 5.3 × 10⁻⁴ |
| 1,351/60 | 22.5166666667 | 6.2 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 507y² = 1. Its smallest solution in positive whole numbers is x = 1,351, y = 60.
√507 in geometry and everyday measurements
- A square garage floor of 507 square feet measures about 22.52 ft (22 ft 6 in) per side, and its corner-to-corner diagonal is √1014 ≈ 31.8 ft.
- 507 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √507 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 11 × 19 box, because 5² + 11² + 19² = 507.
- Since √507 = 13√3, a length of √507 is exactly 13 copies of the length √3 laid end to end.
Square roots near √507 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √504 | 6√14 | 22.4499 | No |
| √505 | √505 | 22.4722 | No |
| √506 | √506 | 22.4944 | No |
| √507 | 13√3 | 22.5167 | No |
| √508 | 2√127 | 22.5389 | No |
| √509 | √509 | 22.5610 | No |
| √510 | √510 | 22.5832 | No |
- The cube root of 507 is about 7.973873.
- Squaring undoes the root: (√507)² = 507, while 507² = 257,049 — the number whose square root is 507.
Frequently asked questions
What is the square root of 507?
The square root of 507 is 13√3 in simplest radical form, which is about 22.5166604984. The negative root, −22.516660, also squares to 507.
Is the square root of 507 rational or irrational?
Irrational. 507 is not a perfect square — it falls between 484 and 529 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √507 be simplified?
Yes. The largest perfect square dividing 507 is 169, so √507 = √169 × √3 = 13√3.
What is √507 rounded to two decimal places?
√507 ≈ 22.52 to two decimal places (22.5 to one, 22.517 to three). Check: 22.52² = 507.1504, close to 507.