√777 at a glance
- Exact value
- √777
- Decimal (10 places)
- 27.8747197295
- Rounded
- 27.9 · 27.87 · 27.875
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.874720
- Prime factorization
- 3 × 7 × 37
- Cube root
- 9.193347
How to simplify √777
The prime factorization of 777 is 3 × 7 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √777 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 777, 3, 7 and 37 appear an odd number of times, so √777 is irrational and 27.8747197295 is a rounded value.
Where √777 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √777 lies between 27 and 28. 777 is 48 above 729 and 7 below 784, so the root is closer to 28.
- Straight line between 729 and 784: 27.8727 (0.01% low)
- Tangent from 27, i.e. 27 + 48 ÷ 54: 27.8889 (0.05% high)
- Tangent from 28, i.e. 28 − 7 ÷ 56: 27.8750 (0% high)
For √777 the tangent at 28 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 777 is just 7 below 784.
Finding √777 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 777: following the tangent line down to zero simplifies to averaging x with 777 ÷ x.
Start from the nearest whole number, 28 (28² = 784):
| Step | Guess x | 777 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 27.7500000000 | 27.8750000000 | 3 |
| 2 | 27.8750000000 | 27.8744394619 | 27.8747197309 | 8 |
| 3 | 27.8747197309 | 27.8747197281 | 27.8747197295 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √777 = 27.8747197295 to every decimal shown.
√777 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √777 the pattern is [27; 1, 6, 1, 54] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √777 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 |
|---|---|---|
| 27/1 | 27.0000000000 | 8.7 × 10⁻¹ |
| 28/1 | 28.0000000000 | 1.3 × 10⁻¹ |
| 195/7 | 27.8571428571 | 1.8 × 10⁻² |
| 223/8 | 27.8750000000 | 2.8 × 10⁻⁴ |
| 12,237/439 | 27.8747152620 | 4.5 × 10⁻⁶ |
| 12,460/447 | 27.8747203579 | 6.3 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 777y² = 1. Its smallest solution in positive whole numbers is x = 223, y = 8.
√777 in geometry and everyday measurements
- 777 square feet is 72.2 m². Laid out as a square — a small house footprint or a lot — it is about 27.87 ft (27 ft 10 in) on a side.
- 777 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √777 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 26 box, because 1² + 10² + 26² = 777.
Square roots near √777 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √774 | 3√86 | 27.8209 | No |
| √775 | 5√31 | 27.8388 | No |
| √776 | 2√194 | 27.8568 | No |
| √777 | √777 | 27.8747 | No |
| √778 | √778 | 27.8927 | No |
| √779 | √779 | 27.9106 | No |
| √780 | 2√195 | 27.9285 | No |
- The cube root of 777 is about 9.193347.
- Squaring undoes the root: (√777)² = 777, while 777² = 603,729 — the number whose square root is 777.
Frequently asked questions
What is the square root of 777?
The square root of 777 is √777, about 27.8747197295. The negative root, −27.874720, also squares to 777.
Is the square root of 777 rational or irrational?
Irrational. 777 is not a perfect square — it falls between 729 and 784 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √777 be simplified?
No. 777 = 3 × 7 × 37 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √777 rounded to two decimal places?
√777 ≈ 27.87 to two decimal places (27.9 to one, 27.875 to three). Check: 27.87² = 776.7369, close to 777.