√558 at a glance
- Exact value
- 3√62
- Decimal (10 places)
- 23.6220236220
- Rounded
- 23.6 · 23.62 · 23.622
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.622024
- Prime factorization
- 2 × 3² × 31
- Cube root
- 8.232746
How to simplify √558
Look for the largest perfect square that divides 558. Here it is 9 (3²), because 558 = 9 × 62 and 62 has no square factor left:
The prime factorization tells the same story: 558 = 2 × 3² × 31. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 31 stays inside.
Check: (3√62)² = 3² × 62 = 9 × 62 = 558. As a decimal, 3√62 = 3 × 7.874007874 ≈ 23.6220236220.
Where √558 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √558 lies between 23 and 24. 558 is 29 above 529 and 18 below 576, so the root is closer to 24.
- Straight line between 529 and 576: 23.6170 (0.02% low)
- Tangent from 23, i.e. 23 + 29 ÷ 46: 23.6304 (0.04% high)
- Tangent from 24, i.e. 24 − 18 ÷ 48: 23.6250 (0.01% high)
For √558 the tangent at 24 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 558 is just 18 below 576.
Finding √558 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, 24 (24² = 576):
| Step | Guess x | 558 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 23.2500000000 | 23.6250000000 | 2 |
| 2 | 23.6250000000 | 23.6190476190 | 23.6220238095 | 6 |
| 3 | 23.6220238095 | 23.6220234345 | 23.6220236220 | 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 √558 = 23.6220236220 to every decimal shown.
√558 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √558 the pattern is [23; 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 46] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √558 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 | 6.2 × 10⁻¹ |
| 24/1 | 24.0000000000 | 3.8 × 10⁻¹ |
| 47/2 | 23.5000000000 | 1.2 × 10⁻¹ |
| 71/3 | 23.6666666667 | 4.5 × 10⁻² |
| 118/5 | 23.6000000000 | 2.2 × 10⁻² |
| 189/8 | 23.6250000000 | 3.0 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 558y² = 1. Its smallest solution in positive whole numbers is x = 7,937, y = 336.
√558 in geometry and everyday measurements
- A square garage floor of 558 square feet measures about 23.62 ft (23 ft 7 in) per side, and its corner-to-corner diagonal is √1116 ≈ 33.4 ft.
- 558 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √558 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 14 × 19 box, because 1² + 14² + 19² = 558.
- Since √558 = 3√62, a length of √558 is exactly 3 copies of the length √62 laid end to end.
Square roots near √558 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √555 | √555 | 23.5584 | No |
| √556 | 2√139 | 23.5797 | No |
| √557 | √557 | 23.6008 | No |
| √558 | 3√62 | 23.6220 | No |
| √559 | √559 | 23.6432 | No |
| √560 | 4√35 | 23.6643 | No |
| √561 | √561 | 23.6854 | No |
- The cube root of 558 is about 8.232746.
- Squaring undoes the root: (√558)² = 558, while 558² = 311,364 — the number whose square root is 558.
Frequently asked questions
What is the square root of 558?
The square root of 558 is 3√62 in simplest radical form, which is about 23.6220236220. The negative root, −23.622024, also squares to 558.
Is the square root of 558 rational or irrational?
Irrational. 558 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 √558 be simplified?
Yes. The largest perfect square dividing 558 is 9, so √558 = √9 × √62 = 3√62.
What is √558 rounded to two decimal places?
√558 ≈ 23.62 to two decimal places (23.6 to one, 23.622 to three). Check: 23.62² = 557.9044, close to 558.