√605 at a glance
- Exact value
- 11√5
- Decimal (10 places)
- 24.5967477525
- Rounded
- 24.6 · 24.60 · 24.597
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.596748
- Prime factorization
- 5 × 11²
- Cube root
- 8.457691
How to simplify √605
Look for the largest perfect square that divides 605. Here it is 121 (11²), because 605 = 121 × 5 and 5 has no square factor left:
The prime factorization tells the same story: 605 = 5 × 11². Each pair of equal primes leaves the radical as one factor, so 11 comes out and 5 stays inside.
Check: (11√5)² = 11² × 5 = 121 × 5 = 605. As a decimal, 11√5 = 11 × 2.2360679775 ≈ 24.5967477525.
Where √605 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √605 lies between 24 and 25. 605 is 29 above 576 and 20 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.5918 (0.02% low)
- Tangent from 24, i.e. 24 + 29 ÷ 48: 24.6042 (0.03% high)
- Tangent from 25, i.e. 25 − 20 ÷ 50: 24.6000 (0.01% high)
For √605 the tangent at 25 wins, missing by only 0.0033. Tangent estimates shine when the number sits close to a perfect square — here 605 is just 20 below 625.
Finding √605 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 605: following the tangent line down to zero simplifies to averaging x with 605 ÷ x.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 605 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.2000000000 | 24.6000000000 | 2 |
| 2 | 24.6000000000 | 24.5934959350 | 24.5967479675 | 6 |
| 3 | 24.5967479675 | 24.5967475375 | 24.5967477525 | 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 √605 = 24.5967477525 to every decimal shown.
√605 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √605 the pattern is [24; 1, 1, 2, 11, 1, 8, 1, 11, 2, 1, 1, 48] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √605 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 | 6.0 × 10⁻¹ |
| 25/1 | 25.0000000000 | 4.0 × 10⁻¹ |
| 49/2 | 24.5000000000 | 9.7 × 10⁻² |
| 123/5 | 24.6000000000 | 3.3 × 10⁻³ |
| 1,402/57 | 24.5964912281 | 2.6 × 10⁻⁴ |
| 1,525/62 | 24.5967741935 | 2.6 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 605y² = 1. Its smallest solution in positive whole numbers is x = 930,249, y = 37,820.
√605 in geometry and everyday measurements
- A square garage floor of 605 square feet measures about 24.6 ft (24 ft 7 in) per side, and its corner-to-corner diagonal is √1210 ≈ 34.8 ft.
- 605 = 11² + 22², so by the Pythagorean theorem √605 is the diagonal of a 11 × 22 rectangle — and the distance between the points (0, 0) and (11, 22) on a grid.
- Since √605 = 11√5, a length of √605 is exactly 11 copies of the length √5 laid end to end.
Square roots near √605 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √602 | √602 | 24.5357 | No |
| √603 | 3√67 | 24.5561 | No |
| √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 |
- The cube root of 605 is about 8.457691.
- Squaring undoes the root: (√605)² = 605, while 605² = 366,025 — the number whose square root is 605.
Frequently asked questions
What is the square root of 605?
The square root of 605 is 11√5 in simplest radical form, which is about 24.5967477525. The negative root, −24.596748, also squares to 605.
Is the square root of 605 rational or irrational?
Irrational. 605 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 √605 be simplified?
Yes. The largest perfect square dividing 605 is 121, so √605 = √121 × √5 = 11√5.
What is √605 rounded to two decimal places?
√605 ≈ 24.60 to two decimal places (24.6 to one, 24.597 to three). Check: 24.60² = 605.16, close to 605.