√427 at a glance
- Exact value
- √427
- Decimal (10 places)
- 20.6639783198
- Rounded
- 20.7 · 20.66 · 20.664
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.663978
- Prime factorization
- 7 × 61
- Cube root
- 7.530248
How to simplify √427
The prime factorization of 427 is 7 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √427 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 427, 7 and 61 appear an odd number of times, so √427 is irrational and 20.6639783198 is a rounded value.
Where √427 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √427 lies between 20 and 21. 427 is 27 above 400 and 14 below 441, so the root is closer to 21.
- Straight line between 400 and 441: 20.6585 (0.03% low)
- Tangent from 20, i.e. 20 + 27 ÷ 40: 20.6750 (0.05% high)
- Tangent from 21, i.e. 21 − 14 ÷ 42: 20.6667 (0.01% high)
For √427 the tangent at 21 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 427 is just 14 below 441.
Finding √427 with the Babylonian method
If a guess is too big, 427 ÷ 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√427) in one step.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 427 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 20.3333333333 | 20.6666666667 | 2 |
| 2 | 20.6666666667 | 20.6612903226 | 20.6639784946 | 6 |
| 3 | 20.6639784946 | 20.6639781449 | 20.6639783198 | 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 √427 = 20.6639783198 to every decimal shown.
√427 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √427 the pattern is [20; 1, 1, 1, 40] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √427 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 |
|---|---|---|
| 20/1 | 20.0000000000 | 6.6 × 10⁻¹ |
| 21/1 | 21.0000000000 | 3.4 × 10⁻¹ |
| 41/2 | 20.5000000000 | 1.6 × 10⁻¹ |
| 62/3 | 20.6666666667 | 2.7 × 10⁻³ |
| 2,521/122 | 20.6639344262 | 4.4 × 10⁻⁵ |
| 2,583/125 | 20.6640000000 | 2.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 427y² = 1. Its smallest solution in positive whole numbers is x = 62, y = 3.
√427 in geometry and everyday measurements
- A square garage floor of 427 square feet measures about 20.66 ft (20 ft 8 in) per side, and its corner-to-corner diagonal is √854 ≈ 29.2 ft.
- 427 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 √427 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 9 × 11 × 15 box, because 9² + 11² + 15² = 427.
Square roots near √427 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √424 | 2√106 | 20.5913 | No |
| √425 | 5√17 | 20.6155 | No |
| √426 | √426 | 20.6398 | No |
| √427 | √427 | 20.6640 | No |
| √428 | 2√107 | 20.6882 | No |
| √429 | √429 | 20.7123 | No |
| √430 | √430 | 20.7364 | No |
- The cube root of 427 is about 7.530248.
- Squaring undoes the root: (√427)² = 427, while 427² = 182,329 — the number whose square root is 427.
Frequently asked questions
What is the square root of 427?
The square root of 427 is √427, about 20.6639783198. The negative root, −20.663978, also squares to 427.
Is the square root of 427 rational or irrational?
Irrational. 427 is not a perfect square — it falls between 400 and 441 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √427 be simplified?
No. 427 = 7 × 61 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √427 rounded to two decimal places?
√427 ≈ 20.66 to two decimal places (20.7 to one, 20.664 to three). Check: 20.66² = 426.8356, close to 427.