√455 at a glance
- Exact value
- √455
- Decimal (10 places)
- 21.3307290077
- Rounded
- 21.3 · 21.33 · 21.331
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.330729
- Prime factorization
- 5 × 7 × 13
- Cube root
- 7.691372
How to simplify √455
The prime factorization of 455 is 5 × 7 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √455 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 455, 5, 7 and 13 appear an odd number of times, so √455 is irrational and 21.3307290077 is a rounded value.
Where √455 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √455 lies between 21 and 22. 455 is 14 above 441 and 29 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.3256 (0.02% low)
- Tangent from 21, i.e. 21 + 14 ÷ 42: 21.3333 (0.01% high)
- Tangent from 22, i.e. 22 − 29 ÷ 44: 21.3409 (0.05% high)
For √455 the tangent at 21 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 455 is just 14 above 441.
Finding √455 with the Babylonian method
If a guess is too big, 455 ÷ 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√455) in one step.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 455 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.6666666667 | 21.3333333333 | 2 |
| 2 | 21.3333333333 | 21.3281250000 | 21.3307291667 | 6 |
| 3 | 21.3307291667 | 21.3307288487 | 21.3307290077 | 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 √455 = 21.3307290077 to every decimal shown.
√455 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √455 the pattern is [21; 3, 42] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √455 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 | 3.3 × 10⁻¹ |
| 64/3 | 21.3333333333 | 2.6 × 10⁻³ |
| 2,709/127 | 21.3307086614 | 2.0 × 10⁻⁵ |
| 8,191/384 | 21.3307291667 | 1.6 × 10⁻⁷ |
| 346,731/16,255 | 21.3307290065 | 1.2 × 10⁻⁹ |
| 1,048,384/49,149 | 21.3307290077 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 455y² = 1. Its smallest solution in positive whole numbers is x = 64, y = 3.
√455 in geometry and everyday measurements
- A square garage floor of 455 square feet measures about 21.33 ft (21 ft 4 in) per side, and its corner-to-corner diagonal is √910 ≈ 30.2 ft.
- 455 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 √455 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √455 as its space diagonal.
Square roots near √455 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √452 | 2√113 | 21.2603 | No |
| √453 | √453 | 21.2838 | No |
| √454 | √454 | 21.3073 | No |
| √455 | √455 | 21.3307 | No |
| √456 | 2√114 | 21.3542 | No |
| √457 | √457 | 21.3776 | No |
| √458 | √458 | 21.4009 | No |
- The cube root of 455 is about 7.691372.
- Squaring undoes the root: (√455)² = 455, while 455² = 207,025 — the number whose square root is 455.
Frequently asked questions
What is the square root of 455?
The square root of 455 is √455, about 21.3307290077. The negative root, −21.330729, also squares to 455.
Is the square root of 455 rational or irrational?
Irrational. 455 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 √455 be simplified?
No. 455 = 5 × 7 × 13 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √455 rounded to two decimal places?
√455 ≈ 21.33 to two decimal places (21.3 to one, 21.331 to three). Check: 21.33² = 454.9689, close to 455.