√27 at a glance
- Exact value
- 3√3
- Decimal (10 places)
- 5.1961524227
- Rounded
- 5.2 · 5.20 · 5.196
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.196152
- Prime factorization
- 3³
- Cube root
- 3
How to simplify √27
Look for the largest perfect square that divides 27. Here it is 9 (3²), because 27 = 9 × 3 and 3 has no square factor left:
The prime factorization tells the same story: 27 = 3³. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 stays inside.
Check: (3√3)² = 3² × 3 = 9 × 3 = 27. As a decimal, 3√3 = 3 × 1.7320508076 ≈ 5.1961524227.
Where √27 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √27 lies between 5 and 6. 27 is 2 above 25 and 9 below 36, so the root is closer to 5.
- Straight line between 25 and 36: 5.1818 (0.28% low)
- Tangent from 5, i.e. 5 + 2 ÷ 10: 5.2000 (0.07% high)
- Tangent from 6, i.e. 6 − 9 ÷ 12: 5.2500 (1.04% high)
For √27 the tangent at 5 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 27 is just 2 above 25.
Finding √27 with the Babylonian method
If a guess is too big, 27 ÷ 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√27) in one step.
Start from the nearest whole number, 5 (5² = 25):
| Step | Guess x | 27 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 5.0000000000 | 5.4000000000 | 5.2000000000 | 2 |
| 2 | 5.2000000000 | 5.1923076923 | 5.1961538462 | 5 |
| 3 | 5.1961538462 | 5.1961509993 | 5.1961524227 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √27 = 5.1961524227 to every decimal shown.
√27 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √27 the pattern is [5; 5, 10] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √27 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 | 2.0 × 10⁻¹ |
| 26/5 | 5.2000000000 | 3.8 × 10⁻³ |
| 265/51 | 5.1960784314 | 7.4 × 10⁻⁵ |
| 1,351/260 | 5.1961538462 | 1.4 × 10⁻⁶ |
| 13,775/2,651 | 5.1961523953 | 2.7 × 10⁻⁸ |
| 70,226/13,515 | 5.1961524232 | 5.3 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 27y² = 1. Its smallest solution in positive whole numbers is x = 26, y = 5.
√27 in geometry and everyday measurements
- A square tile with an area of 27 square inches has sides about 5.196 in long.
- 27 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 √27 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 5 box, because 1² + 1² + 5² = 27.
- Since √27 = 3√3, a length of √27 is exactly 3 copies of the length √3 laid end to end.
Square roots near √27 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √24 | 2√6 | 4.8990 | No |
| √25 | 5 | 5.0000 | Yes |
| √26 | √26 | 5.0990 | No |
| √27 | 3√3 | 5.1962 | No |
| √28 | 2√7 | 5.2915 | No |
| √29 | √29 | 5.3852 | No |
| √30 | √30 | 5.4772 | No |
- The cube root of 27 is exactly 3 — 27 is a perfect cube as well (3³).
- Four times the radicand doubles the root: √108 = 2 × √27 ≈ 10.392305.
Frequently asked questions
What is the square root of 27?
The square root of 27 is 3√3 in simplest radical form, which is about 5.1961524227. The negative root, −5.196152, also squares to 27.
Is the square root of 27 rational or irrational?
Irrational. 27 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 √27 be simplified?
Yes. The largest perfect square dividing 27 is 9, so √27 = √9 × √3 = 3√3.
What is √27 rounded to two decimal places?
√27 ≈ 5.20 to two decimal places (5.2 to one, 5.196 to three). Check: 5.20² = 27.04, close to 27.