√529 at a glance
- Exact value
- 23
- Decimal (10 places)
- 23
- Rounded
- 23
- Perfect square?
- Yes — 23²
- Rational?
- Rational (whole number)
- Both square roots
- ±23
- Prime factorization
- 23²
- Cube root
- 8.087579
How to simplify √529
Write 529 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√529 = 23 = 23
529 has 2 factor pairs (1 × 529, 23 × 23). The pair with two equal numbers, 23 × 23, is what makes it a perfect square — and why it has an odd number of factors.
Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 529 = 1 + 3 + 5 + 7 + … + 45, the first 23 odd numbers.
Where √529 sits between perfect squares
529 is 23². The perfect squares on either side are 484 (22²) and 576 (24²), so the gaps are 45 below and 47 above — consecutive squares always differ by consecutive odd numbers.
Finding √529 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 529: following the tangent line down to zero simplifies to averaging x with 529 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 24:
| Step | Guess x | 529 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 24.0000000000 | 22.0416666667 | 23.0208333333 | 1 |
| 2 | 23.0208333333 | 22.9791855204 | 23.0000094268 | 5 |
| 3 | 23.0000094268 | 22.9999905732 | 23.0000000000 | all 10 shown |
The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √529 = 23.0000000000 to every decimal shown.
√529 in geometry and everyday measurements
- A square garage floor of 529 square feet measures exactly 23 ft (23 ft) per side, and its corner-to-corner diagonal is √1058 ≈ 32.5 ft.
- 529 is not a sum of two whole-number squares — no pair of squares adds up to it — so √529 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 6 × 22 box, because 3² + 6² + 22² = 529.
Square roots near √529 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √531 | 3√59 | 23.0434 | No |
| √532 | 2√133 | 23.0651 | No |
- The cube root of 529 is about 8.087579.
- Taking the square root twice gives the fourth root: √23 ≈ 4.795832.
- Squaring undoes the root: (√529)² = 529, while 529² = 279,841 — the number whose square root is 529.
Frequently asked questions
What is the square root of 529?
The square root of 529 is 23, because 23 × 23 = 529. Strictly, 529 has two square roots, 23 and −23; the √ symbol means the positive one.
Is the square root of 529 rational or irrational?
Rational. 529 is a perfect square, so its square root is the whole number 23.
Can √529 be simplified?
Yes, all the way to a whole number: √529 = 23.
What is √529 rounded to two decimal places?
√529 is exactly 23, or 23.00 written to two decimal places.