√458 at a glance
- Exact value
- √458
- Decimal (10 places)
- 21.4009345590
- Rounded
- 21.4 · 21.40 · 21.401
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.400935
- Prime factorization
- 2 × 229
- Cube root
- 7.708239
How to simplify √458
The prime factorization of 458 is 2 × 229. Every prime appears only once, so there is no pair to bring outside the radical — √458 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 458, 2 and 229 appear an odd number of times, so √458 is irrational and 21.4009345590 is a rounded value.
Where √458 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √458 lies between 21 and 22. 458 is 17 above 441 and 26 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.3953 (0.03% low)
- Tangent from 21, i.e. 21 + 17 ÷ 42: 21.4048 (0.02% high)
- Tangent from 22, i.e. 22 − 26 ÷ 44: 21.4091 (0.04% high)
For √458 the tangent at 21 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 458 is just 17 above 441.
Finding √458 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, 21 (21² = 441):
| Step | Guess x | 458 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.8095238095 | 21.4047619048 | 2 |
| 2 | 21.4047619048 | 21.3971078977 | 21.4009349012 | 6 |
| 3 | 21.4009349012 | 21.4009342169 | 21.4009345590 | 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 √458 = 21.4009345590 to every decimal shown.
√458 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √458 the pattern is [21; 2, 2, 42] with the block of 3 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √458 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 | 4.0 × 10⁻¹ |
| 43/2 | 21.5000000000 | 9.9 × 10⁻² |
| 107/5 | 21.4000000000 | 9.3 × 10⁻⁴ |
| 4,537/212 | 21.4009433962 | 8.8 × 10⁻⁶ |
| 9,181/429 | 21.4009324009 | 2.2 × 10⁻⁶ |
| 22,899/1,070 | 21.4009345794 | 2.0 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 458y² = 1. Its smallest solution in positive whole numbers is x = 22,899, y = 1,070. Because the period is odd, the equation with −1 on the right also has a solution: 107² − 458 × 5² = −1.
√458 in geometry and everyday measurements
- A square garage floor of 458 square feet measures about 21.4 ft (21 ft 5 in) per side, and its corner-to-corner diagonal is √916 ≈ 30.3 ft.
- 458 = 13² + 17², so by the Pythagorean theorem √458 is the diagonal of a 13 × 17 rectangle — and the distance between the points (0, 0) and (13, 17) on a grid.
Square roots near √458 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √455 | √455 | 21.3307 | No |
| √456 | 2√114 | 21.3542 | No |
| √457 | √457 | 21.3776 | No |
| √458 | √458 | 21.4009 | No |
| √459 | 3√51 | 21.4243 | No |
| √460 | 2√115 | 21.4476 | No |
| √461 | √461 | 21.4709 | No |
- The cube root of 458 is about 7.708239.
- Squaring undoes the root: (√458)² = 458, while 458² = 209,764 — the number whose square root is 458.
Frequently asked questions
What is the square root of 458?
The square root of 458 is √458, about 21.4009345590. The negative root, −21.400935, also squares to 458.
Is the square root of 458 rational or irrational?
Irrational. 458 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 √458 be simplified?
No. 458 = 2 × 229 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √458 rounded to two decimal places?
√458 ≈ 21.40 to two decimal places (21.4 to one, 21.401 to three). Check: 21.40² = 457.96, close to 458.