√607 at a glance
- Exact value
- √607
- Decimal (10 places)
- 24.6373699895
- Rounded
- 24.6 · 24.64 · 24.637
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.637370
- Prime factorization
- 607
- Cube root
- 8.467000
How to simplify √607
607 is a prime number, so its only factors are 1 and 607. There is no perfect-square factor to pull out, which means √607 is already in its simplest radical form.
The square root of any prime is irrational. If √607 were a fraction a/b in lowest terms, then a² = 607b², so 607 would divide a — and then 607 would divide b too, contradicting “lowest terms.” That is why the decimal 24.6373699895 is only a rounded value.
Where √607 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √607 lies between 24 and 25. 607 is 31 above 576 and 18 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.6327 (0.02% low)
- Tangent from 24, i.e. 24 + 31 ÷ 48: 24.6458 (0.03% high)
- Tangent from 25, i.e. 25 − 18 ÷ 50: 24.6400 (0.01% high)
For √607 the tangent at 25 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 607 is just 18 below 625.
Finding √607 with the Babylonian method
If a guess is too big, 607 ÷ 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√607) in one step.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 607 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.2800000000 | 24.6400000000 | 2 |
| 2 | 24.6400000000 | 24.6347402597 | 24.6373701299 | 6 |
| 3 | 24.6373701299 | 24.6373698491 | 24.6373699895 | 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 √607 = 24.6373699895 to every decimal shown.
√607 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √607 the pattern is [24; 1, 1, 1, 3, 7, 1, 15, 1, 1, 4, 1, 23, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √607 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 | 6.4 × 10⁻¹ |
| 25/1 | 25.0000000000 | 3.6 × 10⁻¹ |
| 49/2 | 24.5000000000 | 1.4 × 10⁻¹ |
| 74/3 | 24.6666666667 | 2.9 × 10⁻² |
| 271/11 | 24.6363636364 | 1.0 × 10⁻³ |
| 1,971/80 | 24.6375000000 | 1.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 607y² = 1. Its smallest solution in positive whole numbers is x = 164,076,033,968, y = 6,659,640,783.
√607 in geometry and everyday measurements
- A square garage floor of 607 square feet measures about 24.64 ft (24 ft 8 in) per side, and its corner-to-corner diagonal is √1214 ≈ 34.8 ft.
- 607 is not a sum of two whole-number squares — 607 is itself a prime that is one less than a multiple of 4, which rules that out — so √607 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 √607 as its space diagonal.
Square roots near √607 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √604 | 2√151 | 24.5764 | No |
| √605 | 11√5 | 24.5967 | No |
| √606 | √606 | 24.6171 | No |
| √607 | √607 | 24.6374 | No |
| √608 | 4√38 | 24.6577 | No |
| √609 | √609 | 24.6779 | No |
| √610 | √610 | 24.6982 | No |
- The cube root of 607 is about 8.467000.
- Squaring undoes the root: (√607)² = 607, while 607² = 368,449 — the number whose square root is 607.
Frequently asked questions
What is the square root of 607?
The square root of 607 is √607, about 24.6373699895. The negative root, −24.637370, also squares to 607.
Is the square root of 607 rational or irrational?
Irrational. 607 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 √607 be simplified?
No. 607 is prime, so there is no perfect square to take out of the radical.
What is √607 rounded to two decimal places?
√607 ≈ 24.64 to two decimal places (24.6 to one, 24.637 to three). Check: 24.64² = 607.1296, close to 607.