√79 at a glance
- Exact value
- √79
- Decimal (10 places)
- 8.8881944173
- Rounded
- 8.9 · 8.89 · 8.888
- Perfect square?
- No — between 8² and 9²
- Rational?
- Irrational
- Both square roots
- ±8.888194
- Prime factorization
- 79
- Cube root
- 4.290840
How to simplify √79
79 is a prime number, so its only factors are 1 and 79. There is no perfect-square factor to pull out, which means √79 is already in its simplest radical form.
The square root of any prime is irrational. If √79 were a fraction a/b in lowest terms, then a² = 79b², so 79 would divide a — and then 79 would divide b too, contradicting “lowest terms.” That is why the decimal 8.8881944173 is only a rounded value.
Where √79 sits between perfect squares
64 = 8² and 81 = 9² are the nearest perfect squares, so √79 lies between 8 and 9. 79 is 15 above 64 and 2 below 81, so the root is closer to 9.
- Straight line between 64 and 81: 8.8824 (0.07% low)
- Tangent from 8, i.e. 8 + 15 ÷ 16: 8.9375 (0.55% high)
- Tangent from 9, i.e. 9 − 2 ÷ 18: 8.8889 (0.01% high)
For √79 the tangent at 9 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 79 is just 2 below 81.
Finding √79 with the Babylonian method
If a guess is too big, 79 ÷ 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√79) in one step.
Start from the nearest whole number, 9 (9² = 81):
| Step | Guess x | 79 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 9.0000000000 | 8.7777777778 | 8.8888888889 | 3 |
| 2 | 8.8888888889 | 8.8875000000 | 8.8881944444 | 7 |
| 3 | 8.8881944444 | 8.8881943902 | 8.8881944173 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √79 = 8.8881944173 to every decimal shown.
√79 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √79 the pattern is [8; 1, 7, 1, 16] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √79 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 | 8.9 × 10⁻¹ |
| 9/1 | 9.0000000000 | 1.1 × 10⁻¹ |
| 71/8 | 8.8750000000 | 1.3 × 10⁻² |
| 80/9 | 8.8888888889 | 6.9 × 10⁻⁴ |
| 1,351/152 | 8.8881578947 | 3.7 × 10⁻⁵ |
| 1,431/161 | 8.8881987578 | 4.3 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 79y² = 1. Its smallest solution in positive whole numbers is x = 80, y = 9.
√79 in geometry and everyday measurements
- A square room or garden bed covering 79 square feet measures about 8.89 ft (8 ft 11 in) along each wall.
- 79 is not a sum of two whole-number squares — 79 is itself a prime that is one less than a multiple of 4, which rules that out — so √79 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 √79 as its space diagonal.
Square roots near √79 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √82 | √82 | 9.0554 | No |
- The cube root of 79 is about 4.290840.
- Four times the radicand doubles the root: √316 = 2 × √79 ≈ 17.776389.
Frequently asked questions
What is the square root of 79?
The square root of 79 is √79, about 8.8881944173. The negative root, −8.888194, also squares to 79.
Is the square root of 79 rational or irrational?
Irrational. 79 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 √79 be simplified?
No. 79 is prime, so there is no perfect square to take out of the radical.
What is √79 rounded to two decimal places?
√79 ≈ 8.89 to two decimal places (8.9 to one, 8.888 to three). Check: 8.89² = 79.0321, close to 79.