√750 at a glance
- Exact value
- 5√30
- Decimal (10 places)
- 27.3861278753
- Rounded
- 27.4 · 27.39 · 27.386
- Perfect square?
- No — between 27² and 28²
- Rational?
- Irrational
- Both square roots
- ±27.386128
- Prime factorization
- 2 × 3 × 5³
- Cube root
- 9.085603
How to simplify √750
Look for the largest perfect square that divides 750. Here it is 25 (5²), because 750 = 25 × 30 and 30 has no square factor left:
The prime factorization tells the same story: 750 = 2 × 3 × 5³. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 × 3 × 5 stays inside.
Check: (5√30)² = 5² × 30 = 25 × 30 = 750. As a decimal, 5√30 = 5 × 5.4772255751 ≈ 27.3861278753.
Where √750 sits between perfect squares
729 = 27² and 784 = 28² are the nearest perfect squares, so √750 lies between 27 and 28. 750 is 21 above 729 and 34 below 784, so the root is closer to 27.
- Straight line between 729 and 784: 27.3818 (0.02% low)
- Tangent from 27, i.e. 27 + 21 ÷ 54: 27.3889 (0.01% high)
- Tangent from 28, i.e. 28 − 34 ÷ 56: 27.3929 (0.02% high)
For √750 the tangent at 27 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 750 is just 21 above 729.
Finding √750 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 750 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 27.7777777778 | 27.3888888889 | 2 |
| 2 | 27.3888888889 | 27.3833671400 | 27.3861280144 | 6 |
| 3 | 27.3861280144 | 27.3861277361 | 27.3861278753 | 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 √750 = 27.3861278753 to every decimal shown.
√750 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √750 the pattern is [27; 2, 1, 1, 2, 3, 1, 1, 8, 1, 1, 3, 2, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √750 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 | 3.9 × 10⁻¹ |
| 55/2 | 27.5000000000 | 1.1 × 10⁻¹ |
| 82/3 | 27.3333333333 | 5.3 × 10⁻² |
| 137/5 | 27.4000000000 | 1.4 × 10⁻² |
| 356/13 | 27.3846153846 | 1.5 × 10⁻³ |
| 1,205/44 | 27.3863636364 | 2.4 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 750y² = 1. Its smallest solution in positive whole numbers is x = 2,550,251, y = 93,122.
√750 in geometry and everyday measurements
- 750 square feet is 69.7 m². Laid out as a square — a small house footprint or a lot — it is about 27.39 ft (27 ft 5 in) on a side.
- 750 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √750 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 11 × 25 box, because 2² + 11² + 25² = 750.
- Since √750 = 5√30, a length of √750 is exactly 5 copies of the length √30 laid end to end.
Square roots near √750 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √747 | 3√83 | 27.3313 | No |
| √748 | 2√187 | 27.3496 | No |
| √749 | √749 | 27.3679 | No |
| √750 | 5√30 | 27.3861 | No |
| √751 | √751 | 27.4044 | No |
| √752 | 4√47 | 27.4226 | No |
| √753 | √753 | 27.4408 | No |
- The cube root of 750 is about 9.085603.
- Squaring undoes the root: (√750)² = 750, while 750² = 562,500 — the number whose square root is 750.
Frequently asked questions
What is the square root of 750?
The square root of 750 is 5√30 in simplest radical form, which is about 27.3861278753. The negative root, −27.386128, also squares to 750.
Is the square root of 750 rational or irrational?
Irrational. 750 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 √750 be simplified?
Yes. The largest perfect square dividing 750 is 25, so √750 = √25 × √30 = 5√30.
What is √750 rounded to two decimal places?
√750 ≈ 27.39 to two decimal places (27.4 to one, 27.386 to three). Check: 27.39² = 750.2121, close to 750.