√78 at a glance
- Exact value
- √78
- Decimal (10 places)
- 8.8317608663
- Rounded
- 8.8 · 8.83 · 8.832
- Perfect square?
- No — between 8² and 9²
- Rational?
- Irrational
- Both square roots
- ±8.831761
- Prime factorization
- 2 × 3 × 13
- Cube root
- 4.272659
How to simplify √78
The prime factorization of 78 is 2 × 3 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √78 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 78, 2, 3 and 13 appear an odd number of times, so √78 is irrational and 8.8317608663 is a rounded value.
Where √78 sits between perfect squares
64 = 8² and 81 = 9² are the nearest perfect squares, so √78 lies between 8 and 9. 78 is 14 above 64 and 3 below 81, so the root is closer to 9.
- Straight line between 64 and 81: 8.8235 (0.09% low)
- Tangent from 8, i.e. 8 + 14 ÷ 16: 8.8750 (0.49% high)
- Tangent from 9, i.e. 9 − 3 ÷ 18: 8.8333 (0.02% high)
For √78 the tangent at 9 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 78 is just 3 below 81.
Finding √78 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 9 (9² = 81):
| Step | Guess x | 78 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 9.0000000000 | 8.6666666667 | 8.8333333333 | 2 |
| 2 | 8.8333333333 | 8.8301886792 | 8.8317610063 | 6 |
| 3 | 8.8317610063 | 8.8317607264 | 8.8317608663 | 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 √78 = 8.8317608663 to every decimal shown.
√78 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √78 the pattern is [8; 1, 4, 1, 16] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √78 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 | 8.3 × 10⁻¹ |
| 9/1 | 9.0000000000 | 1.7 × 10⁻¹ |
| 44/5 | 8.8000000000 | 3.2 × 10⁻² |
| 53/6 | 8.8333333333 | 1.6 × 10⁻³ |
| 892/101 | 8.8316831683 | 7.8 × 10⁻⁵ |
| 945/107 | 8.8317757009 | 1.5 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 78y² = 1. Its smallest solution in positive whole numbers is x = 53, y = 6.
√78 in geometry and everyday measurements
- A square room or garden bed covering 78 square feet measures about 8.83 ft (8 ft 10 in) along each wall.
- 78 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 √78 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 7 box, because 2² + 5² + 7² = 78.
Square roots near √78 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √81 | 9 | 9.0000 | Yes |
- The cube root of 78 is about 4.272659.
- Four times the radicand doubles the root: √312 = 2 × √78 ≈ 17.663522.
Frequently asked questions
What is the square root of 78?
The square root of 78 is √78, about 8.8317608663. The negative root, −8.831761, also squares to 78.
Is the square root of 78 rational or irrational?
Irrational. 78 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 √78 be simplified?
No. 78 = 2 × 3 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √78 rounded to two decimal places?
√78 ≈ 8.83 to two decimal places (8.8 to one, 8.832 to three). Check: 8.83² = 77.9689, close to 78.