√567 at a glance
- Exact value
- 9√7
- Decimal (10 places)
- 23.8117617996
- Rounded
- 23.8 · 23.81 · 23.812
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.811762
- Prime factorization
- 3⁴ × 7
- Cube root
- 8.276773
How to simplify √567
Look for the largest perfect square that divides 567. Here it is 81 (9²), because 567 = 81 × 7 and 7 has no square factor left:
The prime factorization tells the same story: 567 = 3⁴ × 7. Each pair of equal primes leaves the radical as one factor, so 3² comes out and 7 stays inside.
567 has 2 square factors (9 and 81). Starting with a smaller one still works but takes more rounds: √567 = 3√63, and √63 can be simplified again. Using 81 straight away finishes in one step.
Check: (9√7)² = 9² × 7 = 81 × 7 = 567. As a decimal, 9√7 = 9 × 2.6457513111 ≈ 23.8117617996.
Where √567 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √567 lies between 23 and 24. 567 is 38 above 529 and 9 below 576, so the root is closer to 24.
- Straight line between 529 and 576: 23.8085 (0.01% low)
- Tangent from 23, i.e. 23 + 38 ÷ 46: 23.8261 (0.06% high)
- Tangent from 24, i.e. 24 − 9 ÷ 48: 23.8125 (0% high)
For √567 the tangent at 24 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 567 is just 9 below 576.
Finding √567 with the Babylonian method
If a guess is too big, 567 ÷ 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√567) in one step.
Start from the nearest whole number, 24 (24² = 576):
| Step | Guess x | 567 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 23.6250000000 | 23.8125000000 | 3 |
| 2 | 23.8125000000 | 23.8110236220 | 23.8117618110 | 7 |
| 3 | 23.8117618110 | 23.8117617881 | 23.8117617996 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √567 = 23.8117617996 to every decimal shown.
√567 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √567 the pattern is [23; 1, 4, 3, 4, 1, 46] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √567 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 | 8.1 × 10⁻¹ |
| 24/1 | 24.0000000000 | 1.9 × 10⁻¹ |
| 119/5 | 23.8000000000 | 1.2 × 10⁻² |
| 381/16 | 23.8125000000 | 7.4 × 10⁻⁴ |
| 1,643/69 | 23.8115942029 | 1.7 × 10⁻⁴ |
| 2,024/85 | 23.8117647059 | 2.9 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 567y² = 1. Its smallest solution in positive whole numbers is x = 2,024, y = 85.
√567 in geometry and everyday measurements
- A square garage floor of 567 square feet measures about 23.81 ft (23 ft 10 in) per side, and its corner-to-corner diagonal is √1134 ≈ 33.7 ft.
- 567 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 √567 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 √567 as its space diagonal.
- Since √567 = 9√7, a length of √567 is exactly 9 copies of the length √7 laid end to end.
Square roots near √567 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √564 | 2√141 | 23.7487 | No |
| √565 | √565 | 23.7697 | No |
| √566 | √566 | 23.7908 | No |
| √567 | 9√7 | 23.8118 | No |
| √568 | 2√142 | 23.8328 | No |
| √569 | √569 | 23.8537 | No |
| √570 | √570 | 23.8747 | No |
- The cube root of 567 is about 8.276773.
- Squaring undoes the root: (√567)² = 567, while 567² = 321,489 — the number whose square root is 567.
Frequently asked questions
What is the square root of 567?
The square root of 567 is 9√7 in simplest radical form, which is about 23.8117617996. The negative root, −23.811762, also squares to 567.
Is the square root of 567 rational or irrational?
Irrational. 567 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 √567 be simplified?
Yes. The largest perfect square dividing 567 is 81, so √567 = √81 × √7 = 9√7.
What is √567 rounded to two decimal places?
√567 ≈ 23.81 to two decimal places (23.8 to one, 23.812 to three). Check: 23.81² = 566.9161, close to 567.