√179 at a glance
- Exact value
- √179
- Decimal (10 places)
- 13.3790881603
- Rounded
- 13.4 · 13.38 · 13.379
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.379088
- Prime factorization
- 179
- Cube root
- 5.635741
How to simplify √179
179 is a prime number, so its only factors are 1 and 179. There is no perfect-square factor to pull out, which means √179 is already in its simplest radical form.
The square root of any prime is irrational. If √179 were a fraction a/b in lowest terms, then a² = 179b², so 179 would divide a — and then 179 would divide b too, contradicting “lowest terms.” That is why the decimal 13.3790881603 is only a rounded value.
Where √179 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √179 lies between 13 and 14. 179 is 10 above 169 and 17 below 196, so the root is closer to 13.
- Straight line between 169 and 196: 13.3704 (0.07% low)
- Tangent from 13, i.e. 13 + 10 ÷ 26: 13.3846 (0.04% high)
- Tangent from 14, i.e. 14 − 17 ÷ 28: 13.3929 (0.1% high)
For √179 the tangent at 13 wins, missing by only 0.0055. Tangent estimates shine when the number sits close to a perfect square — here 179 is just 10 above 169.
Finding √179 with the Babylonian method
If a guess is too big, 179 ÷ 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√179) in one step.
Start from the nearest whole number, 13 (13² = 169):
| Step | Guess x | 179 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 13.7692307692 | 13.3846153846 | 2 |
| 2 | 13.3846153846 | 13.3735632184 | 13.3790893015 | 5 |
| 3 | 13.3790893015 | 13.3790870190 | 13.3790881603 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √179 = 13.3790881603 to every decimal shown.
√179 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √179 the pattern is [13; 2, 1, 1, 1, 3, 5, 13, 5, 3, 1, 1, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √179 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 |
|---|---|---|
| 13/1 | 13.0000000000 | 3.8 × 10⁻¹ |
| 27/2 | 13.5000000000 | 1.2 × 10⁻¹ |
| 40/3 | 13.3333333333 | 4.6 × 10⁻² |
| 67/5 | 13.4000000000 | 2.1 × 10⁻² |
| 107/8 | 13.3750000000 | 4.1 × 10⁻³ |
| 388/29 | 13.3793103448 | 2.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 179y² = 1. Its smallest solution in positive whole numbers is x = 4,190,210, y = 313,191.
√179 in geometry and everyday measurements
- A square room or garden bed covering 179 square feet measures about 13.38 ft (13 ft 5 in) along each wall.
- 179 is not a sum of two whole-number squares — 179 is itself a prime that is one less than a multiple of 4, which rules that out — so √179 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 13 box, because 1² + 3² + 13² = 179.
Square roots near √179 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √176 | 4√11 | 13.2665 | No |
| √177 | √177 | 13.3041 | No |
| √178 | √178 | 13.3417 | No |
| √179 | √179 | 13.3791 | No |
| √180 | 6√5 | 13.4164 | No |
| √181 | √181 | 13.4536 | No |
| √182 | √182 | 13.4907 | No |
- The cube root of 179 is about 5.635741.
- Four times the radicand doubles the root: √716 = 2 × √179 ≈ 26.758176.
Frequently asked questions
What is the square root of 179?
The square root of 179 is √179, about 13.3790881603. The negative root, −13.379088, also squares to 179.
Is the square root of 179 rational or irrational?
Irrational. 179 is not a perfect square — it falls between 169 and 196 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √179 be simplified?
No. 179 is prime, so there is no perfect square to take out of the radical.
What is √179 rounded to two decimal places?
√179 ≈ 13.38 to two decimal places (13.4 to one, 13.379 to three). Check: 13.38² = 179.0244, close to 179.