√77 at a glance
- Exact value
- √77
- Decimal (10 places)
- 8.7749643874
- Rounded
- 8.8 · 8.77 · 8.775
- Perfect square?
- No — between 8² and 9²
- Rational?
- Irrational
- Both square roots
- ±8.774964
- Prime factorization
- 7 × 11
- Cube root
- 4.254321
How to simplify √77
The prime factorization of 77 is 7 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √77 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 77, 7 and 11 appear an odd number of times, so √77 is irrational and 8.7749643874 is a rounded value.
Where √77 sits between perfect squares
64 = 8² and 81 = 9² are the nearest perfect squares, so √77 lies between 8 and 9. 77 is 13 above 64 and 4 below 81, so the root is closer to 9.
- Straight line between 64 and 81: 8.7647 (0.12% low)
- Tangent from 8, i.e. 8 + 13 ÷ 16: 8.8125 (0.43% high)
- Tangent from 9, i.e. 9 − 4 ÷ 18: 8.7778 (0.03% high)
For √77 the tangent at 9 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 77 is just 4 below 81.
Finding √77 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 77: following the tangent line down to zero simplifies to averaging x with 77 ÷ x.
Start from the nearest whole number, 9 (9² = 81):
| Step | Guess x | 77 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 9.0000000000 | 8.5555555556 | 8.7777777778 | 2 |
| 2 | 8.7777777778 | 8.7721518987 | 8.7749648383 | 6 |
| 3 | 8.7749648383 | 8.7749639365 | 8.7749643874 | 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 √77 = 8.7749643874 to every decimal shown.
√77 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √77 the pattern is [8; 1, 3, 2, 3, 1, 16] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √77 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 |
|---|---|---|
| 8/1 | 8.0000000000 | 7.7 × 10⁻¹ |
| 9/1 | 9.0000000000 | 2.3 × 10⁻¹ |
| 35/4 | 8.7500000000 | 2.5 × 10⁻² |
| 79/9 | 8.7777777778 | 2.8 × 10⁻³ |
| 272/31 | 8.7741935484 | 7.7 × 10⁻⁴ |
| 351/40 | 8.7750000000 | 3.6 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 77y² = 1. Its smallest solution in positive whole numbers is x = 351, y = 40.
√77 in geometry and everyday measurements
- A square room or garden bed covering 77 square feet measures about 8.77 ft (8 ft 9 in) along each wall.
- 77 is not a sum of two whole-number squares — the prime factor 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √77 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 8 box, because 2² + 3² + 8² = 77.
Square roots near √77 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √74 | √74 | 8.6023 | No |
| √75 | 5√3 | 8.6603 | No |
| √76 | 2√19 | 8.7178 | No |
| √77 | √77 | 8.7750 | No |
| √78 | √78 | 8.8318 | No |
| √79 | √79 | 8.8882 | No |
| √80 | 4√5 | 8.9443 | No |
- The cube root of 77 is about 4.254321.
- Four times the radicand doubles the root: √308 = 2 × √77 ≈ 17.549929.
Frequently asked questions
What is the square root of 77?
The square root of 77 is √77, about 8.7749643874. The negative root, −8.774964, also squares to 77.
Is the square root of 77 rational or irrational?
Irrational. 77 is not a perfect square — it falls between 64 and 81 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √77 be simplified?
No. 77 = 7 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √77 rounded to two decimal places?
√77 ≈ 8.77 to two decimal places (8.8 to one, 8.775 to three). Check: 8.77² = 76.9129, close to 77.