√30 at a glance
- Exact value
- √30
- Decimal (10 places)
- 5.4772255751
- Rounded
- 5.5 · 5.48 · 5.477
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.477226
- Prime factorization
- 2 × 3 × 5
- Cube root
- 3.107233
How to simplify √30
The prime factorization of 30 is 2 × 3 × 5. Every prime appears only once, so there is no pair to bring outside the radical — √30 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 30, 2, 3 and 5 appear an odd number of times, so √30 is irrational and 5.4772255751 is a rounded value.
Where √30 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √30 lies between 5 and 6. 30 is 5 above 25 and 6 below 36, so the root is closer to 5.
- Straight line between 25 and 36: 5.4545 (0.41% low)
- Tangent from 5, i.e. 5 + 5 ÷ 10: 5.5000 (0.42% high)
- Tangent from 6, i.e. 6 − 6 ÷ 12: 5.5000 (0.42% high)
For √30 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.
Finding √30 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, 5 (5² = 25):
| Step | Guess x | 30 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 5.0000000000 | 6.0000000000 | 5.5000000000 | 1 |
| 2 | 5.5000000000 | 5.4545454545 | 5.4772727273 | 4 |
| 3 | 5.4772727273 | 5.4771784232 | 5.4772255753 | 9 |
| 4 | 5.4772255753 | 5.4772255748 | 5.4772255751 | all 10 shown |
The count of correct decimals went 1, 4, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √30 = 5.4772255751 to every decimal shown.
√30 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √30 the pattern is [5; 2, 10] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √30 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 |
|---|---|---|
| 5/1 | 5.0000000000 | 4.8 × 10⁻¹ |
| 11/2 | 5.5000000000 | 2.3 × 10⁻² |
| 115/21 | 5.4761904762 | 1.0 × 10⁻³ |
| 241/44 | 5.4772727273 | 4.7 × 10⁻⁵ |
| 2,525/461 | 5.4772234273 | 2.1 × 10⁻⁶ |
| 5,291/966 | 5.4772256729 | 9.8 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 30y² = 1. Its smallest solution in positive whole numbers is x = 11, y = 2.
√30 in geometry and everyday measurements
- A square tile with an area of 30 square inches has sides about 5.477 in long.
- 30 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 √30 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 5 box, because 1² + 2² + 5² = 30.
Square roots near √30 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √27 | 3√3 | 5.1962 | No |
| √28 | 2√7 | 5.2915 | No |
| √29 | √29 | 5.3852 | No |
| √30 | √30 | 5.4772 | No |
| √31 | √31 | 5.5678 | No |
| √32 | 4√2 | 5.6569 | No |
| √33 | √33 | 5.7446 | No |
- The cube root of 30 is about 3.107233.
- Four times the radicand doubles the root: √120 = 2 × √30 ≈ 10.954451.
Frequently asked questions
What is the square root of 30?
The square root of 30 is √30, about 5.4772255751. The negative root, −5.477226, also squares to 30.
Is the square root of 30 rational or irrational?
Irrational. 30 is not a perfect square — it falls between 25 and 36 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √30 be simplified?
No. 30 = 2 × 3 × 5 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √30 rounded to two decimal places?
√30 ≈ 5.48 to two decimal places (5.5 to one, 5.477 to three). Check: 5.48² = 30.0304, close to 30.