√465 at a glance
- Exact value
- √465
- Decimal (10 places)
- 21.5638586528
- Rounded
- 21.6 · 21.56 · 21.564
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.563859
- Prime factorization
- 3 × 5 × 31
- Cube root
- 7.747311
How to simplify √465
The prime factorization of 465 is 3 × 5 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √465 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 465, 3, 5 and 31 appear an odd number of times, so √465 is irrational and 21.5638586528 is a rounded value.
Where √465 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √465 lies between 21 and 22. 465 is 24 above 441 and 19 below 484, so the root is closer to 22.
- Straight line between 441 and 484: 21.5581 (0.03% low)
- Tangent from 21, i.e. 21 + 24 ÷ 42: 21.5714 (0.04% high)
- Tangent from 22, i.e. 22 − 19 ÷ 44: 21.5682 (0.02% high)
For √465 the tangent at 22 wins, missing by only 0.0043. Tangent estimates shine when the number sits close to a perfect square — here 465 is just 19 below 484.
Finding √465 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 465: following the tangent line down to zero simplifies to averaging x with 465 ÷ x.
Start from the nearest whole number, 22 (22² = 484):
| Step | Guess x | 465 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 21.1363636364 | 21.5681818182 | 2 |
| 2 | 21.5681818182 | 21.5595363541 | 21.5638590861 | 6 |
| 3 | 21.5638590861 | 21.5638582196 | 21.5638586528 | 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 √465 = 21.5638586528 to every decimal shown.
√465 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √465 the pattern is [21; 1, 1, 3, 2, 2, 2, 3, 1, 1, 42] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √465 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 5.6 × 10⁻¹ |
| 22/1 | 22.0000000000 | 4.4 × 10⁻¹ |
| 43/2 | 21.5000000000 | 6.4 × 10⁻² |
| 151/7 | 21.5714285714 | 7.6 × 10⁻³ |
| 345/16 | 21.5625000000 | 1.4 × 10⁻³ |
| 841/39 | 21.5641025641 | 2.4 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 465y² = 1. Its smallest solution in positive whole numbers is x = 15,871, y = 736.
√465 in geometry and everyday measurements
- A square garage floor of 465 square feet measures about 21.56 ft (21 ft 7 in) per side, and its corner-to-corner diagonal is √930 ≈ 30.5 ft.
- 465 is not a sum of two whole-number squares — the prime factor 3 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √465 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 20 box, because 1² + 8² + 20² = 465.
Square roots near √465 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √462 | √462 | 21.4942 | No |
| √463 | √463 | 21.5174 | No |
| √464 | 4√29 | 21.5407 | No |
| √465 | √465 | 21.5639 | No |
| √466 | √466 | 21.5870 | No |
| √467 | √467 | 21.6102 | No |
| √468 | 6√13 | 21.6333 | No |
- The cube root of 465 is about 7.747311.
- Squaring undoes the root: (√465)² = 465, while 465² = 216,225 — the number whose square root is 465.
Frequently asked questions
What is the square root of 465?
The square root of 465 is √465, about 21.5638586528. The negative root, −21.563859, also squares to 465.
Is the square root of 465 rational or irrational?
Irrational. 465 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √465 be simplified?
No. 465 = 3 × 5 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √465 rounded to two decimal places?
√465 ≈ 21.56 to two decimal places (21.6 to one, 21.564 to three). Check: 21.56² = 464.8336, close to 465.