√624 at a glance
- Exact value
- 4√39
- Decimal (10 places)
- 24.9799919936
- Rounded
- 25.0 · 24.98 · 24.980
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.979992
- Prime factorization
- 2⁴ × 3 × 13
- Cube root
- 8.545317
How to simplify √624
Look for the largest perfect square that divides 624. Here it is 16 (4²), because 624 = 16 × 39 and 39 has no square factor left:
The prime factorization tells the same story: 624 = 2⁴ × 3 × 13. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 3 × 13 stays inside.
624 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √624 = 2√156, and √156 can be simplified again. Using 16 straight away finishes in one step.
Check: (4√39)² = 4² × 39 = 16 × 39 = 624. As a decimal, 4√39 = 4 × 6.2449979984 ≈ 24.9799919936.
Where √624 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √624 lies between 24 and 25. 624 is 48 above 576 and 1 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.9796 (0% low)
- Tangent from 24, i.e. 24 + 48 ÷ 48: 25.0000 (0.08% high)
- Tangent from 25, i.e. 25 − 1 ÷ 50: 24.9800 (0% high)
For √624 the tangent at 25 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 624 is just 1 below 625.
Finding √624 with the Babylonian method
Picture a rectangle with an area of 624 and one side x; the other side must be 624 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √624.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 624 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.9600000000 | 24.9800000000 | 5 |
| 2 | 24.9800000000 | 24.9799839872 | 24.9799919936 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √624 = 24.9799919936 to every decimal shown.
√624 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √624 the pattern is [24; 1, 48] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √624 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 | 9.8 × 10⁻¹ |
| 25/1 | 25.0000000000 | 2.0 × 10⁻² |
| 1,224/49 | 24.9795918367 | 4.0 × 10⁻⁴ |
| 1,249/50 | 24.9800000000 | 8.0 × 10⁻⁶ |
| 61,176/2,449 | 24.9799918334 | 1.6 × 10⁻⁷ |
| 62,425/2,499 | 24.9799919968 | 3.2 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 624y² = 1. Its smallest solution in positive whole numbers is x = 25, y = 1.
√624 in geometry and everyday measurements
- A square garage floor of 624 square feet measures about 24.98 ft (25 ft) per side, and its corner-to-corner diagonal is √1248 ≈ 35.3 ft.
- 624 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 √624 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √624 as its space diagonal.
- Since √624 = 4√39, a length of √624 is exactly 4 copies of the length √39 laid end to end.
Square roots near √624 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √621 | 3√69 | 24.9199 | No |
| √622 | √622 | 24.9399 | No |
| √623 | √623 | 24.9600 | No |
| √624 | 4√39 | 24.9800 | No |
| √625 | 25 | 25.0000 | Yes |
| √626 | √626 | 25.0200 | No |
| √627 | √627 | 25.0400 | No |
- The cube root of 624 is about 8.545317.
- Because 624 = 4 × 156, the root is twice √156: 2 × 12.489996 ≈ 24.979992.
Frequently asked questions
What is the square root of 624?
The square root of 624 is 4√39 in simplest radical form, which is about 24.9799919936. The negative root, −24.979992, also squares to 624.
Is the square root of 624 rational or irrational?
Irrational. 624 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 √624 be simplified?
Yes. The largest perfect square dividing 624 is 16, so √624 = √16 × √39 = 4√39.
What is √624 rounded to two decimal places?
√624 ≈ 24.98 to two decimal places (25.0 to one, 24.980 to three). Check: 24.98² = 624.0004, close to 624.