√729 at a glance
- Exact value
- 27
- Decimal (10 places)
- 27
- Rounded
- 27
- Perfect square?
- Yes — 27²
- Rational?
- Rational (whole number)
- Both square roots
- ±27
- Prime factorization
- 3⁶
- Cube root
- 9
How to simplify √729
Write 729 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√729 = 3³ = 27
729 has 4 factor pairs (1 × 729, 3 × 243, 9 × 81, 27 × 27). The pair with two equal numbers, 27 × 27, 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 729 = 1 + 3 + 5 + 7 + … + 53, the first 27 odd numbers.
Where √729 sits between perfect squares
729 is 27². The perfect squares on either side are 676 (26²) and 784 (28²), so the gaps are 53 below and 55 above — consecutive squares always differ by consecutive odd numbers.
Finding √729 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 729: following the tangent line down to zero simplifies to averaging x with 729 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 28:
| Step | Guess x | 729 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 28.0000000000 | 26.0357142857 | 27.0178571429 | 1 |
| 2 | 27.0178571429 | 26.9821546596 | 27.0000059012 | 5 |
| 3 | 27.0000059012 | 26.9999940988 | 27.0000000000 | 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 √729 = 27.0000000000 to every decimal shown.
√729 in geometry and everyday measurements
- 729 square feet is 67.7 m². Laid out as a square — a small house footprint or a lot — it is exactly 27 ft (27 ft) on a side.
- 729 is not a sum of two whole-number squares — no pair of squares adds up to it — so √729 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 7 × 26 box, because 2² + 7² + 26² = 729.
Square roots near √729 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √726 | 11√6 | 26.9444 | No |
| √727 | √727 | 26.9629 | No |
| √728 | 2√182 | 26.9815 | No |
| √729 | 27 | 27.0000 | Yes |
| √730 | √730 | 27.0185 | No |
| √731 | √731 | 27.0370 | No |
| √732 | 2√183 | 27.0555 | No |
- The cube root of 729 is exactly 9 — 729 is a perfect cube as well (9³), which makes it a perfect sixth power (3⁶).
- Taking the square root twice gives the fourth root: √27 ≈ 5.196152.
- Squaring undoes the root: (√729)² = 729, while 729² = 531,441 — the number whose square root is 729.
Frequently asked questions
What is the square root of 729?
The square root of 729 is 27, because 27 × 27 = 729. Strictly, 729 has two square roots, 27 and −27; the √ symbol means the positive one.
Is the square root of 729 rational or irrational?
Rational. 729 is a perfect square, so its square root is the whole number 27.
Can √729 be simplified?
Yes, all the way to a whole number: √729 = 27.
What is √729 rounded to two decimal places?
√729 is exactly 27, or 27.00 written to two decimal places.