√727 at a glance
- Exact value
- √727
- Decimal (10 places)
- 26.9629375254
- Rounded
- 27.0 · 26.96 · 26.963
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.962938
- Prime factorization
- 727
- Cube root
- 8.991762
How to simplify √727
727 is a prime number, so its only factors are 1 and 727. There is no perfect-square factor to pull out, which means √727 is already in its simplest radical form.
The square root of any prime is irrational. If √727 were a fraction a/b in lowest terms, then a² = 727b², so 727 would divide a — and then 727 would divide b too, contradicting “lowest terms.” That is why the decimal 26.9629375254 is only a rounded value.
Where √727 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √727 lies between 26 and 27. 727 is 51 above 676 and 2 below 729, so the root is closer to 27.
- Straight line between 676 and 729: 26.9623 (0% low)
- Tangent from 26, i.e. 26 + 51 ÷ 52: 26.9808 (0.07% high)
- Tangent from 27, i.e. 27 − 2 ÷ 54: 26.9630 (0% high)
For √727 the tangent at 27 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 727 is just 2 below 729.
Finding √727 with the Babylonian method
If a guess is too big, 727 ÷ 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√727) in one step.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 727 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 26.9259259259 | 26.9629629630 | 4 |
| 2 | 26.9629629630 | 26.9629120879 | 26.9629375254 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √727 = 26.9629375254 to every decimal shown.
√727 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √727 the pattern is [26; 1, 25, 1, 52] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √727 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 |
|---|---|---|
| 26/1 | 26.0000000000 | 9.6 × 10⁻¹ |
| 27/1 | 27.0000000000 | 3.7 × 10⁻² |
| 701/26 | 26.9615384615 | 1.4 × 10⁻³ |
| 728/27 | 26.9629629630 | 2.5 × 10⁻⁵ |
| 38,557/1,430 | 26.9629370629 | 4.6 × 10⁻⁷ |
| 39,285/1,457 | 26.9629375429 | 1.7 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 727y² = 1. Its smallest solution in positive whole numbers is x = 728, y = 27.
√727 in geometry and everyday measurements
- 727 square feet is 67.5 m². Laid out as a square — a small house footprint or a lot — it is about 26.96 ft (27 ft) on a side.
- 727 is not a sum of two whole-number squares — 727 is itself a prime that is one less than a multiple of 4, which rules that out — so √727 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 √727 as its space diagonal.
Square roots near √727 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √724 | 2√181 | 26.9072 | No |
| √725 | 5√29 | 26.9258 | No |
| √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 |
- The cube root of 727 is about 8.991762.
- Squaring undoes the root: (√727)² = 727, while 727² = 528,529 — the number whose square root is 727.
Frequently asked questions
What is the square root of 727?
The square root of 727 is √727, about 26.9629375254. The negative root, −26.962938, also squares to 727.
Is the square root of 727 rational or irrational?
Irrational. 727 is not a perfect square — it falls between 676 and 729 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √727 be simplified?
No. 727 is prime, so there is no perfect square to take out of the radical.
What is √727 rounded to two decimal places?
√727 ≈ 26.96 to two decimal places (27.0 to one, 26.963 to three). Check: 26.96² = 726.8416, close to 727.