√594 at a glance
- Exact value
- 3√66
- Decimal (10 places)
- 24.3721152139
- Rounded
- 24.4 · 24.37 · 24.372
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.372115
- Prime factorization
- 2 × 3³ × 11
- Cube root
- 8.406118
How to simplify √594
Look for the largest perfect square that divides 594. Here it is 9 (3²), because 594 = 9 × 66 and 66 has no square factor left:
The prime factorization tells the same story: 594 = 2 × 3³ × 11. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 3 × 11 stays inside.
Check: (3√66)² = 3² × 66 = 9 × 66 = 594. As a decimal, 3√66 = 3 × 8.1240384046 ≈ 24.3721152139.
Where √594 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √594 lies between 24 and 25. 594 is 18 above 576 and 31 below 625, so the root is closer to 24.
- Straight line between 576 and 625: 24.3673 (0.02% low)
- Tangent from 24, i.e. 24 + 18 ÷ 48: 24.3750 (0.01% high)
- Tangent from 25, i.e. 25 − 31 ÷ 50: 24.3800 (0.03% high)
For √594 the tangent at 24 wins, missing by only 0.0029. Tangent estimates shine when the number sits close to a perfect square — here 594 is just 18 above 576.
Finding √594 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 | 594 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 24.7500000000 | 24.3750000000 | 2 |
| 2 | 24.3750000000 | 24.3692307692 | 24.3721153846 | 6 |
| 3 | 24.3721153846 | 24.3721150432 | 24.3721152139 | 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 √594 = 24.3721152139 to every decimal shown.
√594 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √594 the pattern is [24; 2, 1, 2, 5, 24, 5, 2, 1, 2, 48] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √594 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 |
|---|---|---|
| 24/1 | 24.0000000000 | 3.7 × 10⁻¹ |
| 49/2 | 24.5000000000 | 1.3 × 10⁻¹ |
| 73/3 | 24.3333333333 | 3.9 × 10⁻² |
| 195/8 | 24.3750000000 | 2.9 × 10⁻³ |
| 1,048/43 | 24.3720930233 | 2.2 × 10⁻⁵ |
| 25,347/1,040 | 24.3721153846 | 1.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 594y² = 1. Its smallest solution in positive whole numbers is x = 1,098,305, y = 45,064.
√594 in geometry and everyday measurements
- A square garage floor of 594 square feet measures about 24.37 ft (24 ft 4 in) per side, and its corner-to-corner diagonal is √1188 ≈ 34.5 ft.
- 594 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √594 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 23 box, because 1² + 8² + 23² = 594.
- Since √594 = 3√66, a length of √594 is exactly 3 copies of the length √66 laid end to end.
Square roots near √594 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √591 | √591 | 24.3105 | No |
| √592 | 4√37 | 24.3311 | No |
| √593 | √593 | 24.3516 | No |
| √594 | 3√66 | 24.3721 | No |
| √595 | √595 | 24.3926 | No |
| √596 | 2√149 | 24.4131 | No |
| √597 | √597 | 24.4336 | No |
- The cube root of 594 is about 8.406118.
- Squaring undoes the root: (√594)² = 594, while 594² = 352,836 — the number whose square root is 594.
Frequently asked questions
What is the square root of 594?
The square root of 594 is 3√66 in simplest radical form, which is about 24.3721152139. The negative root, −24.372115, also squares to 594.
Is the square root of 594 rational or irrational?
Irrational. 594 is not a perfect square — it falls between 576 and 625 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √594 be simplified?
Yes. The largest perfect square dividing 594 is 9, so √594 = √9 × √66 = 3√66.
What is √594 rounded to two decimal places?
√594 ≈ 24.37 to two decimal places (24.4 to one, 24.372 to three). Check: 24.37² = 593.8969, close to 594.