√755 at a glance
- Exact value
- √755
- Decimal (10 places)
- 27.4772633281
- Rounded
- 27.5 · 27.48 · 27.477
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.477263
- Prime factorization
- 5 × 151
- Cube root
- 9.105748
How to simplify √755
The prime factorization of 755 is 5 × 151. Every prime appears only once, so there is no pair to bring outside the radical — √755 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 755, 5 and 151 appear an odd number of times, so √755 is irrational and 27.4772633281 is a rounded value.
Where √755 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √755 lies between 27 and 28. 755 is 26 above 729 and 29 below 784, so the root is closer to 27.
- Straight line between 729 and 784: 27.4727 (0.02% low)
- Tangent from 27, i.e. 27 + 26 ÷ 54: 27.4815 (0.02% high)
- Tangent from 28, i.e. 28 − 29 ÷ 56: 27.4821 (0.02% high)
For √755 the tangent at 27 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 755 is just 26 above 729.
Finding √755 with the Babylonian method
If a guess is too big, 755 ÷ 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√755) in one step.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 755 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 27.9629629630 | 27.4814814815 | 2 |
| 2 | 27.4814814815 | 27.4730458221 | 27.4772636518 | 6 |
| 3 | 27.4772636518 | 27.4772630043 | 27.4772633281 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √755 = 27.4772633281 to every decimal shown.
√755 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √755 the pattern is [27; 2, 10, 2, 54] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √755 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 |
|---|---|---|
| 27/1 | 27.0000000000 | 4.8 × 10⁻¹ |
| 55/2 | 27.5000000000 | 2.3 × 10⁻² |
| 577/21 | 27.4761904762 | 1.1 × 10⁻³ |
| 1,209/44 | 27.4772727273 | 9.4 × 10⁻⁶ |
| 65,863/2,397 | 27.4772632457 | 8.2 × 10⁻⁸ |
| 132,935/4,838 | 27.4772633320 | 3.9 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 755y² = 1. Its smallest solution in positive whole numbers is x = 1,209, y = 44.
√755 in geometry and everyday measurements
- 755 square feet is 70.1 m². Laid out as a square — a small house footprint or a lot — it is about 27.48 ft (27 ft 6 in) on a side.
- 755 is not a sum of two whole-number squares — the prime factor 151 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √755 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 27 box, because 1² + 5² + 27² = 755.
Square roots near √755 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √752 | 4√47 | 27.4226 | No |
| √753 | √753 | 27.4408 | No |
| √754 | √754 | 27.4591 | No |
| √755 | √755 | 27.4773 | No |
| √756 | 6√21 | 27.4955 | No |
| √757 | √757 | 27.5136 | No |
| √758 | √758 | 27.5318 | No |
- The cube root of 755 is about 9.105748.
- Squaring undoes the root: (√755)² = 755, while 755² = 570,025 — the number whose square root is 755.
Frequently asked questions
What is the square root of 755?
The square root of 755 is √755, about 27.4772633281. The negative root, −27.477263, also squares to 755.
Is the square root of 755 rational or irrational?
Irrational. 755 is not a perfect square — it falls between 729 and 784 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √755 be simplified?
No. 755 = 5 × 151 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √755 rounded to two decimal places?
√755 ≈ 27.48 to two decimal places (27.5 to one, 27.477 to three). Check: 27.48² = 755.1504, close to 755.