√451 at a glance
- Exact value
- √451
- Decimal (10 places)
- 21.2367605816
- Rounded
- 21.2 · 21.24 · 21.237
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.236761
- Prime factorization
- 11 × 41
- Cube root
- 7.668766
How to simplify √451
The prime factorization of 451 is 11 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √451 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 451, 11 and 41 appear an odd number of times, so √451 is irrational and 21.2367605816 is a rounded value.
Where √451 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √451 lies between 21 and 22. 451 is 10 above 441 and 33 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.2326 (0.02% low)
- Tangent from 21, i.e. 21 + 10 ÷ 42: 21.2381 (0.01% high)
- Tangent from 22, i.e. 22 − 33 ÷ 44: 21.2500 (0.06% high)
For √451 the tangent at 21 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 451 is just 10 above 441.
Finding √451 with the Babylonian method
If a guess is too big, 451 ÷ 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√451) in one step.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 451 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.4761904762 | 21.2380952381 | 2 |
| 2 | 21.2380952381 | 21.2354260090 | 21.2367606235 | 7 |
| 3 | 21.2367606235 | 21.2367605397 | 21.2367605816 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √451 = 21.2367605816 to every decimal shown.
√451 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √451 the pattern is [21; 4, 4, 2, 8, 21, 8, 2, 4, 4, 42] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √451 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 | 2.4 × 10⁻¹ |
| 85/4 | 21.2500000000 | 1.3 × 10⁻² |
| 361/17 | 21.2352941176 | 1.5 × 10⁻³ |
| 807/38 | 21.2368421053 | 8.2 × 10⁻⁵ |
| 6,817/321 | 21.2367601246 | 4.6 × 10⁻⁷ |
| 143,964/6,779 | 21.2367605842 | 2.6 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 451y² = 1. Its smallest solution in positive whole numbers is x = 46,471,490, y = 2,188,257.
√451 in geometry and everyday measurements
- A square garage floor of 451 square feet measures about 21.24 ft (21 ft 3 in) per side, and its corner-to-corner diagonal is √902 ≈ 30 ft.
- 451 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √451 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 21 box, because 1² + 3² + 21² = 451.
Square roots near √451 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √448 | 8√7 | 21.1660 | No |
| √449 | √449 | 21.1896 | No |
| √450 | 15√2 | 21.2132 | No |
| √451 | √451 | 21.2368 | No |
| √452 | 2√113 | 21.2603 | No |
| √453 | √453 | 21.2838 | No |
| √454 | √454 | 21.3073 | No |
- The cube root of 451 is about 7.668766.
- Squaring undoes the root: (√451)² = 451, while 451² = 203,401 — the number whose square root is 451.
Frequently asked questions
What is the square root of 451?
The square root of 451 is √451, about 21.2367605816. The negative root, −21.236761, also squares to 451.
Is the square root of 451 rational or irrational?
Irrational. 451 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 √451 be simplified?
No. 451 = 11 × 41 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √451 rounded to two decimal places?
√451 ≈ 21.24 to two decimal places (21.2 to one, 21.237 to three). Check: 21.24² = 451.1376, close to 451.