√182 at a glance
- Exact value
- √182
- Decimal (10 places)
- 13.4907375632
- Rounded
- 13.5 · 13.49 · 13.491
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.490738
- Prime factorization
- 2 × 7 × 13
- Cube root
- 5.667051
How to simplify √182
The prime factorization of 182 is 2 × 7 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √182 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 182, 2, 7 and 13 appear an odd number of times, so √182 is irrational and 13.4907375632 is a rounded value.
Where √182 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √182 lies between 13 and 14. 182 is 13 above 169 and 14 below 196, so the root is closer to 13.
- Straight line between 169 and 196: 13.4815 (0.07% low)
- Tangent from 13, i.e. 13 + 13 ÷ 26: 13.5000 (0.07% high)
- Tangent from 14, i.e. 14 − 14 ÷ 28: 13.5000 (0.07% high)
For √182 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.
Finding √182 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, 13 (13² = 169):
| Step | Guess x | 182 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 14.0000000000 | 13.5000000000 | 2 |
| 2 | 13.5000000000 | 13.4814814815 | 13.4907407407 | 5 |
| 3 | 13.4907407407 | 13.4907343857 | 13.4907375632 | 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 √182 = 13.4907375632 to every decimal shown.
√182 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √182 the pattern is [13; 2, 26] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √182 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 | 4.9 × 10⁻¹ |
| 27/2 | 13.5000000000 | 9.3 × 10⁻³ |
| 715/53 | 13.4905660377 | 1.7 × 10⁻⁴ |
| 1,457/108 | 13.4907407407 | 3.2 × 10⁻⁶ |
| 38,597/2,861 | 13.4907375044 | 5.9 × 10⁻⁸ |
| 78,651/5,830 | 13.4907375643 | 1.1 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 182y² = 1. Its smallest solution in positive whole numbers is x = 27, y = 2.
√182 in geometry and everyday measurements
- A square room or garden bed covering 182 square feet measures about 13.49 ft (13 ft 6 in) along each wall.
- 182 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √182 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 10 box, because 1² + 9² + 10² = 182.
Square roots near √182 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √179 | √179 | 13.3791 | No |
| √180 | 6√5 | 13.4164 | No |
| √181 | √181 | 13.4536 | No |
| √182 | √182 | 13.4907 | No |
| √183 | √183 | 13.5277 | No |
| √184 | 2√46 | 13.5647 | No |
| √185 | √185 | 13.6015 | No |
- The cube root of 182 is about 5.667051.
- Four times the radicand doubles the root: √728 = 2 × √182 ≈ 26.981475.
Frequently asked questions
What is the square root of 182?
The square root of 182 is √182, about 13.4907375632. The negative root, −13.490738, also squares to 182.
Is the square root of 182 rational or irrational?
Irrational. 182 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 √182 be simplified?
No. 182 = 2 × 7 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √182 rounded to two decimal places?
√182 ≈ 13.49 to two decimal places (13.5 to one, 13.491 to three). Check: 13.49² = 181.9801, close to 182.