√615 at a glance
- Exact value
- √615
- Decimal (10 places)
- 24.7991935353
- Rounded
- 24.8 · 24.80 · 24.799
- Perfect square?
- No — between 24² and 25²
- Rational?
- Irrational
- Both square roots
- ±24.799194
- Prime factorization
- 3 × 5 × 41
- Cube root
- 8.504035
How to simplify √615
The prime factorization of 615 is 3 × 5 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √615 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 615, 3, 5 and 41 appear an odd number of times, so √615 is irrational and 24.7991935353 is a rounded value.
Where √615 sits between perfect squares
576 = 24² and 625 = 25² are the nearest perfect squares, so √615 lies between 24 and 25. 615 is 39 above 576 and 10 below 625, so the root is closer to 25.
- Straight line between 576 and 625: 24.7959 (0.01% low)
- Tangent from 24, i.e. 24 + 39 ÷ 48: 24.8125 (0.05% high)
- Tangent from 25, i.e. 25 − 10 ÷ 50: 24.8000 (0% high)
For √615 the tangent at 25 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 615 is just 10 below 625.
Finding √615 with the Babylonian method
If a guess is too big, 615 ÷ 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√615) in one step.
Start from the nearest whole number, 25 (25² = 625):
| Step | Guess x | 615 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 25.0000000000 | 24.6000000000 | 24.8000000000 | 3 |
| 2 | 24.8000000000 | 24.7983870968 | 24.7991935484 | 7 |
| 3 | 24.7991935484 | 24.7991935222 | 24.7991935353 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √615 = 24.7991935353 to every decimal shown.
√615 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √615 the pattern is [24; 1, 3, 1, 48] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √615 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 | 8.0 × 10⁻¹ |
| 25/1 | 25.0000000000 | 2.0 × 10⁻¹ |
| 99/4 | 24.7500000000 | 4.9 × 10⁻² |
| 124/5 | 24.8000000000 | 8.1 × 10⁻⁴ |
| 6,051/244 | 24.7991803279 | 1.3 × 10⁻⁵ |
| 6,175/249 | 24.7991967871 | 3.3 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 615y² = 1. Its smallest solution in positive whole numbers is x = 124, y = 5.
√615 in geometry and everyday measurements
- A square garage floor of 615 square feet measures about 24.8 ft (24 ft 10 in) per side, and its corner-to-corner diagonal is √1230 ≈ 35.1 ft.
- 615 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 √615 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 √615 as its space diagonal.
Square roots near √615 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √612 | 6√17 | 24.7386 | No |
| √613 | √613 | 24.7588 | No |
| √614 | √614 | 24.7790 | No |
| √615 | √615 | 24.7992 | No |
| √616 | 2√154 | 24.8193 | No |
| √617 | √617 | 24.8395 | No |
| √618 | √618 | 24.8596 | No |
- The cube root of 615 is about 8.504035.
- Squaring undoes the root: (√615)² = 615, while 615² = 378,225 — the number whose square root is 615.
Frequently asked questions
What is the square root of 615?
The square root of 615 is √615, about 24.7991935353. The negative root, −24.799194, also squares to 615.
Is the square root of 615 rational or irrational?
Irrational. 615 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 √615 be simplified?
No. 615 = 3 × 5 × 41 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √615 rounded to two decimal places?
√615 ≈ 24.80 to two decimal places (24.8 to one, 24.799 to three). Check: 24.80² = 615.04, close to 615.