√753 at a glance
- Exact value
- √753
- Decimal (10 places)
- 27.4408454680
- Rounded
- 27.4 · 27.44 · 27.441
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.440845
- Prime factorization
- 3 × 251
- Cube root
- 9.097701
How to simplify √753
The prime factorization of 753 is 3 × 251. Every prime appears only once, so there is no pair to bring outside the radical — √753 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 753, 3 and 251 appear an odd number of times, so √753 is irrational and 27.4408454680 is a rounded value.
Where √753 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √753 lies between 27 and 28. 753 is 24 above 729 and 31 below 784, so the root is closer to 27.
- Straight line between 729 and 784: 27.4364 (0.02% low)
- Tangent from 27, i.e. 27 + 24 ÷ 54: 27.4444 (0.01% high)
- Tangent from 28, i.e. 28 − 31 ÷ 56: 27.4464 (0.02% high)
For √753 the tangent at 27 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 753 is just 24 above 729.
Finding √753 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 753: following the tangent line down to zero simplifies to averaging x with 753 ÷ x.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 753 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 27.8888888889 | 27.4444444444 | 2 |
| 2 | 27.4444444444 | 27.4372469636 | 27.4408457040 | 6 |
| 3 | 27.4408457040 | 27.4408452320 | 27.4408454680 | 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 √753 = 27.4408454680 to every decimal shown.
√753 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √753 the pattern is [27; 2, 3, 1, 2, 1, 1, 1, 7, 4, 1, 6, 18, …] with the block of 24 terms after the semicolon repeating forever (only the first 12 of the 24 are shown). A pattern that never ends is one more proof that √753 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 | 4.4 × 10⁻¹ |
| 55/2 | 27.5000000000 | 5.9 × 10⁻² |
| 192/7 | 27.4285714286 | 1.2 × 10⁻² |
| 247/9 | 27.4444444444 | 3.6 × 10⁻³ |
| 686/25 | 27.4400000000 | 8.5 × 10⁻⁴ |
| 933/34 | 27.4411764706 | 3.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 753y² = 1. Its smallest solution in positive whole numbers is x = 308,526,027,863, y = 11,243,313,484.
√753 in geometry and everyday measurements
- 753 square feet is 70 m². Laid out as a square — a small house footprint or a lot — it is about 27.44 ft (27 ft 5 in) on a side.
- 753 is not a sum of two whole-number squares — the prime factor 3 and 251 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √753 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 8 × 25 box, because 8² + 8² + 25² = 753.
Square roots near √753 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √750 | 5√30 | 27.3861 | No |
| √751 | √751 | 27.4044 | No |
| √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 |
- The cube root of 753 is about 9.097701.
- Squaring undoes the root: (√753)² = 753, while 753² = 567,009 — the number whose square root is 753.
Frequently asked questions
What is the square root of 753?
The square root of 753 is √753, about 27.4408454680. The negative root, −27.440845, also squares to 753.
Is the square root of 753 rational or irrational?
Irrational. 753 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 √753 be simplified?
No. 753 = 3 × 251 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √753 rounded to two decimal places?
√753 ≈ 27.44 to two decimal places (27.4 to one, 27.441 to three). Check: 27.44² = 752.9536, close to 753.