√787 at a glance
- Exact value
- √787
- Decimal (10 places)
- 28.0535202782
- Rounded
- 28.1 · 28.05 · 28.054
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.053520
- Prime factorization
- 787
- Cube root
- 9.232619
How to simplify √787
787 is a prime number, so its only factors are 1 and 787. There is no perfect-square factor to pull out, which means √787 is already in its simplest radical form.
The square root of any prime is irrational. If √787 were a fraction a/b in lowest terms, then a² = 787b², so 787 would divide a — and then 787 would divide b too, contradicting “lowest terms.” That is why the decimal 28.0535202782 is only a rounded value.
Where √787 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √787 lies between 28 and 29. 787 is 3 above 784 and 54 below 841, so the root is closer to 28.
- Straight line between 784 and 841: 28.0526 (0% low)
- Tangent from 28, i.e. 28 + 3 ÷ 56: 28.0536 (0% high)
- Tangent from 29, i.e. 29 − 54 ÷ 58: 28.0690 (0.06% high)
For √787 the tangent at 28 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 787 is just 3 above 784.
Finding √787 with the Babylonian method
If a guess is too big, 787 ÷ 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√787) in one step.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 787 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 28.1071428571 | 28.0535714286 | 4 |
| 2 | 28.0535714286 | 28.0534691279 | 28.0535202783 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √787 = 28.0535202782 to every decimal shown.
√787 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √787 the pattern is [28; 18, 1, 2, 5, 1, 8, 1, 1, 27, 1, 1, 8, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √787 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 | 5.4 × 10⁻² |
| 505/18 | 28.0555555556 | 2.0 × 10⁻³ |
| 533/19 | 28.0526315789 | 8.9 × 10⁻⁴ |
| 1,571/56 | 28.0535714286 | 5.1 × 10⁻⁵ |
| 8,388/299 | 28.0535117057 | 8.6 × 10⁻⁶ |
| 9,959/355 | 28.0535211268 | 8.5 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 787y² = 1. Its smallest solution in positive whole numbers is x = 34,625,394,242, y = 1,234,262,007.
√787 in geometry and everyday measurements
- 787 square feet is 73.1 m². Laid out as a square — a small house footprint or a lot — it is about 28.05 ft (28 ft 1 in) on a side.
- 787 is not a sum of two whole-number squares — 787 is itself a prime that is one less than a multiple of 4, which rules that out — so √787 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 7 × 27 box, because 3² + 7² + 27² = 787.
Square roots near √787 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √784 | 28 | 28.0000 | Yes |
| √785 | √785 | 28.0179 | No |
| √786 | √786 | 28.0357 | No |
| √787 | √787 | 28.0535 | No |
| √788 | 2√197 | 28.0713 | No |
| √789 | √789 | 28.0891 | No |
| √790 | √790 | 28.1069 | No |
- The cube root of 787 is about 9.232619.
- Squaring undoes the root: (√787)² = 787, while 787² = 619,369 — the number whose square root is 787.
Frequently asked questions
What is the square root of 787?
The square root of 787 is √787, about 28.0535202782. The negative root, −28.053520, also squares to 787.
Is the square root of 787 rational or irrational?
Irrational. 787 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 √787 be simplified?
No. 787 is prime, so there is no perfect square to take out of the radical.
What is √787 rounded to two decimal places?
√787 ≈ 28.05 to two decimal places (28.1 to one, 28.054 to three). Check: 28.05² = 786.8025, close to 787.