√527 at a glance
- Exact value
- √527
- Decimal (10 places)
- 22.9564805665
- Rounded
- 23.0 · 22.96 · 22.956
- Perfect square?
- No — between 22² and 23²
- Rational?
- Irrational
- Both square roots
- ±22.956481
- Prime factorization
- 17 × 31
- Cube root
- 8.077374
How to simplify √527
The prime factorization of 527 is 17 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √527 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 527, 17 and 31 appear an odd number of times, so √527 is irrational and 22.9564805665 is a rounded value.
Where √527 sits between perfect squares
484 = 22² and 529 = 23² are the nearest perfect squares, so √527 lies between 22 and 23. 527 is 43 above 484 and 2 below 529, so the root is closer to 23.
- Straight line between 484 and 529: 22.9556 (0% low)
- Tangent from 22, i.e. 22 + 43 ÷ 44: 22.9773 (0.09% high)
- Tangent from 23, i.e. 23 − 2 ÷ 46: 22.9565 (0% high)
For √527 the tangent at 23 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 527 is just 2 below 529.
Finding √527 with the Babylonian method
If a guess is too big, 527 ÷ 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√527) in one step.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 527 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 22.9130434783 | 22.9565217391 | 4 |
| 2 | 22.9565217391 | 22.9564393939 | 22.9564805665 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √527 = 22.9564805665 to every decimal shown.
√527 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √527 the pattern is [22; 1, 21, 1, 44] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √527 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 |
|---|---|---|
| 22/1 | 22.0000000000 | 9.6 × 10⁻¹ |
| 23/1 | 23.0000000000 | 4.4 × 10⁻² |
| 505/22 | 22.9545454545 | 1.9 × 10⁻³ |
| 528/23 | 22.9565217391 | 4.1 × 10⁻⁵ |
| 23,737/1,034 | 22.9564796905 | 8.8 × 10⁻⁷ |
| 24,265/1,057 | 22.9564806055 | 3.9 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 527y² = 1. Its smallest solution in positive whole numbers is x = 528, y = 23.
√527 in geometry and everyday measurements
- A square garage floor of 527 square feet measures about 22.96 ft (22 ft 11 in) per side, and its corner-to-corner diagonal is √1054 ≈ 32.5 ft.
- 527 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √527 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 √527 as its space diagonal.
Square roots near √527 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √524 | 2√131 | 22.8910 | No |
| √525 | 5√21 | 22.9129 | No |
| √526 | √526 | 22.9347 | No |
| √527 | √527 | 22.9565 | No |
| √528 | 4√33 | 22.9783 | No |
| √529 | 23 | 23.0000 | Yes |
| √530 | √530 | 23.0217 | No |
- The cube root of 527 is about 8.077374.
- Squaring undoes the root: (√527)² = 527, while 527² = 277,729 — the number whose square root is 527.
Frequently asked questions
What is the square root of 527?
The square root of 527 is √527, about 22.9564805665. The negative root, −22.956481, also squares to 527.
Is the square root of 527 rational or irrational?
Irrational. 527 is not a perfect square — it falls between 484 and 529 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √527 be simplified?
No. 527 = 17 × 31 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √527 rounded to two decimal places?
√527 ≈ 22.96 to two decimal places (23.0 to one, 22.956 to three). Check: 22.96² = 527.1616, close to 527.