√165 at a glance
- Exact value
- √165
- Decimal (10 places)
- 12.8452325787
- Rounded
- 12.8 · 12.85 · 12.845
- Perfect square?
- No — between 12² and 13²
- Rational?
- Irrational
- Both square roots
- ±12.845233
- Prime factorization
- 3 × 5 × 11
- Cube root
- 5.484807
How to simplify √165
The prime factorization of 165 is 3 × 5 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √165 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 165, 3, 5 and 11 appear an odd number of times, so √165 is irrational and 12.8452325787 is a rounded value.
Where √165 sits between perfect squares
144 = 12² and 169 = 13² are the nearest perfect squares, so √165 lies between 12 and 13. 165 is 21 above 144 and 4 below 169, so the root is closer to 13.
- Straight line between 144 and 169: 12.8400 (0.04% low)
- Tangent from 12, i.e. 12 + 21 ÷ 24: 12.8750 (0.23% high)
- Tangent from 13, i.e. 13 − 4 ÷ 26: 12.8462 (0.01% high)
For √165 the tangent at 13 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 165 is just 4 below 169.
Finding √165 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 165: following the tangent line down to zero simplifies to averaging x with 165 ÷ x.
Start from the nearest whole number, 13 (13² = 169):
| Step | Guess x | 165 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 12.6923076923 | 12.8461538462 | 3 |
| 2 | 12.8461538462 | 12.8443113772 | 12.8452326117 | 7 |
| 3 | 12.8452326117 | 12.8452325456 | 12.8452325787 | 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 √165 = 12.8452325787 to every decimal shown.
√165 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √165 the pattern is [12; 1, 5, 2, 5, 1, 24] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √165 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 | 8.5 × 10⁻¹ |
| 13/1 | 13.0000000000 | 1.5 × 10⁻¹ |
| 77/6 | 12.8333333333 | 1.2 × 10⁻² |
| 167/13 | 12.8461538462 | 9.2 × 10⁻⁴ |
| 912/71 | 12.8450704225 | 1.6 × 10⁻⁴ |
| 1,079/84 | 12.8452380952 | 5.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 165y² = 1. Its smallest solution in positive whole numbers is x = 1,079, y = 84.
√165 in geometry and everyday measurements
- A square room or garden bed covering 165 square feet measures about 12.85 ft (12 ft 10 in) along each wall.
- 165 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √165 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 10 box, because 1² + 8² + 10² = 165.
Square roots near √165 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √162 | 9√2 | 12.7279 | No |
| √163 | √163 | 12.7671 | No |
| √164 | 2√41 | 12.8062 | No |
| √165 | √165 | 12.8452 | No |
| √166 | √166 | 12.8841 | No |
| √167 | √167 | 12.9228 | No |
| √168 | 2√42 | 12.9615 | No |
- The cube root of 165 is about 5.484807.
- Four times the radicand doubles the root: √660 = 2 × √165 ≈ 25.690465.
Frequently asked questions
What is the square root of 165?
The square root of 165 is √165, about 12.8452325787. The negative root, −12.845233, also squares to 165.
Is the square root of 165 rational or irrational?
Irrational. 165 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 √165 be simplified?
No. 165 = 3 × 5 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √165 rounded to two decimal places?
√165 ≈ 12.85 to two decimal places (12.8 to one, 12.845 to three). Check: 12.85² = 165.1225, close to 165.