√815 at a glance
- Exact value
- √815
- Decimal (10 places)
- 28.5482048472
- Rounded
- 28.5 · 28.55 · 28.548
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.548205
- Prime factorization
- 5 × 163
- Cube root
- 9.340839
How to simplify √815
The prime factorization of 815 is 5 × 163. Every prime appears only once, so there is no pair to bring outside the radical — √815 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 815, 5 and 163 appear an odd number of times, so √815 is irrational and 28.5482048472 is a rounded value.
Where √815 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √815 lies between 28 and 29. 815 is 31 above 784 and 26 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.5439 (0.02% low)
- Tangent from 28, i.e. 28 + 31 ÷ 56: 28.5536 (0.02% high)
- Tangent from 29, i.e. 29 − 26 ÷ 58: 28.5517 (0.01% high)
For √815 the tangent at 29 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 815 is just 26 below 841.
Finding √815 with the Babylonian method
If a guess is too big, 815 ÷ 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√815) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 815 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.1034482759 | 28.5517241379 | 2 |
| 2 | 28.5517241379 | 28.5446859903 | 28.5482050641 | 6 |
| 3 | 28.5482050641 | 28.5482046303 | 28.5482048472 | 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 √815 = 28.5482048472 to every decimal shown.
√815 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √815 the pattern is [28; 1, 1, 4, 1, 2, 5, 2, 1, 4, 1, 1, 56] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √815 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 5.5 × 10⁻¹ |
| 29/1 | 29.0000000000 | 4.5 × 10⁻¹ |
| 57/2 | 28.5000000000 | 4.8 × 10⁻² |
| 257/9 | 28.5555555556 | 7.4 × 10⁻³ |
| 314/11 | 28.5454545455 | 2.8 × 10⁻³ |
| 885/31 | 28.5483870968 | 1.8 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 815y² = 1. Its smallest solution in positive whole numbers is x = 156,644, y = 5,487.
√815 in geometry and everyday measurements
- 815 square feet is 75.7 m². Laid out as a square — a small house footprint or a lot — it is about 28.55 ft (28 ft 7 in) on a side.
- 815 is not a sum of two whole-number squares — the prime factor 163 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √815 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 √815 as its space diagonal.
Square roots near √815 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √812 | 2√203 | 28.4956 | No |
| √813 | √813 | 28.5132 | No |
| √814 | √814 | 28.5307 | No |
| √815 | √815 | 28.5482 | No |
| √816 | 4√51 | 28.5657 | No |
| √817 | √817 | 28.5832 | No |
| √818 | √818 | 28.6007 | No |
- The cube root of 815 is about 9.340839.
- Squaring undoes the root: (√815)² = 815, while 815² = 664,225 — the number whose square root is 815.
Frequently asked questions
What is the square root of 815?
The square root of 815 is √815, about 28.5482048472. The negative root, −28.548205, also squares to 815.
Is the square root of 815 rational or irrational?
Irrational. 815 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √815 be simplified?
No. 815 = 5 × 163 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √815 rounded to two decimal places?
√815 ≈ 28.55 to two decimal places (28.5 to one, 28.548 to three). Check: 28.55² = 815.1025, close to 815.