√562 at a glance
- Exact value
- √562
- Decimal (10 places)
- 23.7065391823
- Rounded
- 23.7 · 23.71 · 23.707
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.706539
- Prime factorization
- 2 × 281
- Cube root
- 8.252372
How to simplify √562
The prime factorization of 562 is 2 × 281. Every prime appears only once, so there is no pair to bring outside the radical — √562 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 562, 2 and 281 appear an odd number of times, so √562 is irrational and 23.7065391823 is a rounded value.
Where √562 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √562 lies between 23 and 24. 562 is 33 above 529 and 14 below 576, so the root is closer to 24.
- Straight line between 529 and 576: 23.7021 (0.02% low)
- Tangent from 23, i.e. 23 + 33 ÷ 46: 23.7174 (0.05% high)
- Tangent from 24, i.e. 24 − 14 ÷ 48: 23.7083 (0.01% high)
For √562 the tangent at 24 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 562 is just 14 below 576.
Finding √562 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, 24 (24² = 576):
| Step | Guess x | 562 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 23.4166666667 | 23.7083333333 | 2 |
| 2 | 23.7083333333 | 23.7047451670 | 23.7065392501 | 7 |
| 3 | 23.7065392501 | 23.7065391144 | 23.7065391823 | 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 √562 = 23.7065391823 to every decimal shown.
√562 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √562 the pattern is [23; 1, 2, 2, 2, 4, 1, 5, 1, 22, 1, 5, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √562 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 |
|---|---|---|
| 23/1 | 23.0000000000 | 7.1 × 10⁻¹ |
| 24/1 | 24.0000000000 | 2.9 × 10⁻¹ |
| 71/3 | 23.6666666667 | 4.0 × 10⁻² |
| 166/7 | 23.7142857143 | 7.7 × 10⁻³ |
| 403/17 | 23.7058823529 | 6.6 × 10⁻⁴ |
| 1,778/75 | 23.7066666667 | 1.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 562y² = 1. Its smallest solution in positive whole numbers is x = 220,938,497, y = 9,319,728.
√562 in geometry and everyday measurements
- A square garage floor of 562 square feet measures about 23.71 ft (23 ft 8 in) per side, and its corner-to-corner diagonal is √1124 ≈ 33.5 ft.
- 562 = 11² + 21², so by the Pythagorean theorem √562 is the diagonal of a 11 × 21 rectangle — and the distance between the points (0, 0) and (11, 21) on a grid.
Square roots near √562 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √559 | √559 | 23.6432 | No |
| √560 | 4√35 | 23.6643 | No |
| √561 | √561 | 23.6854 | No |
| √562 | √562 | 23.7065 | No |
| √563 | √563 | 23.7276 | No |
| √564 | 2√141 | 23.7487 | No |
| √565 | √565 | 23.7697 | No |
- The cube root of 562 is about 8.252372.
- Squaring undoes the root: (√562)² = 562, while 562² = 315,844 — the number whose square root is 562.
Frequently asked questions
What is the square root of 562?
The square root of 562 is √562, about 23.7065391823. The negative root, −23.706539, also squares to 562.
Is the square root of 562 rational or irrational?
Irrational. 562 is not a perfect square — it falls between 529 and 576 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √562 be simplified?
No. 562 = 2 × 281 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √562 rounded to two decimal places?
√562 ≈ 23.71 to two decimal places (23.7 to one, 23.707 to three). Check: 23.71² = 562.1641, close to 562.