√161 at a glance
- Exact value
- √161
- Decimal (10 places)
- 12.6885775404
- Rounded
- 12.7 · 12.69 · 12.689
- Perfect square?
- No — between 12² and 13²
- Rational?
- Irrational
- Both square roots
- ±12.688578
- Prime factorization
- 7 × 23
- Cube root
- 5.440122
How to simplify √161
The prime factorization of 161 is 7 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √161 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 161, 7 and 23 appear an odd number of times, so √161 is irrational and 12.6885775404 is a rounded value.
Where √161 sits between perfect squares
144 = 12² and 169 = 13² are the nearest perfect squares, so √161 lies between 12 and 13. 161 is 17 above 144 and 8 below 169, so the root is closer to 13.
- Straight line between 144 and 169: 12.6800 (0.07% low)
- Tangent from 12, i.e. 12 + 17 ÷ 24: 12.7083 (0.16% high)
- Tangent from 13, i.e. 13 − 8 ÷ 26: 12.6923 (0.03% high)
For √161 the tangent at 13 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 161 is just 8 below 169.
Finding √161 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 161: following the tangent line down to zero simplifies to averaging x with 161 ÷ x.
Start from the nearest whole number, 13 (13² = 169):
| Step | Guess x | 161 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 12.3846153846 | 12.6923076923 | 2 |
| 2 | 12.6923076923 | 12.6848484848 | 12.6885780886 | 6 |
| 3 | 12.6885780886 | 12.6885769923 | 12.6885775404 | 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 √161 = 12.6885775404 to every decimal shown.
√161 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √161 the pattern is [12; 1, 2, 4, 1, 2, 1, 4, 2, 1, 24] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √161 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 |
|---|---|---|
| 12/1 | 12.0000000000 | 6.9 × 10⁻¹ |
| 13/1 | 13.0000000000 | 3.1 × 10⁻¹ |
| 38/3 | 12.6666666667 | 2.2 × 10⁻² |
| 165/13 | 12.6923076923 | 3.7 × 10⁻³ |
| 203/16 | 12.6875000000 | 1.1 × 10⁻³ |
| 571/45 | 12.6888888889 | 3.1 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 161y² = 1. Its smallest solution in positive whole numbers is x = 11,775, y = 928.
√161 in geometry and everyday measurements
- A square room or garden bed covering 161 square feet measures about 12.69 ft (12 ft 8 in) along each wall.
- 161 is not a sum of two whole-number squares — the prime factor 7 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √161 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 12 box, because 1² + 4² + 12² = 161.
Square roots near √161 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √158 | √158 | 12.5698 | No |
| √159 | √159 | 12.6095 | No |
| √160 | 4√10 | 12.6491 | No |
| √161 | √161 | 12.6886 | No |
| √162 | 9√2 | 12.7279 | No |
| √163 | √163 | 12.7671 | No |
| √164 | 2√41 | 12.8062 | No |
- The cube root of 161 is about 5.440122.
- Four times the radicand doubles the root: √644 = 2 × √161 ≈ 25.377155.
Frequently asked questions
What is the square root of 161?
The square root of 161 is √161, about 12.6885775404. The negative root, −12.688578, also squares to 161.
Is the square root of 161 rational or irrational?
Irrational. 161 is not a perfect square — it falls between 144 and 169 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √161 be simplified?
No. 161 = 7 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √161 rounded to two decimal places?
√161 ≈ 12.69 to two decimal places (12.7 to one, 12.689 to three). Check: 12.69² = 161.0361, close to 161.