√55 at a glance
- Exact value
- √55
- Decimal (10 places)
- 7.4161984871
- Rounded
- 7.4 · 7.42 · 7.416
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.416198
- Prime factorization
- 5 × 11
- Cube root
- 3.802952
How to simplify √55
The prime factorization of 55 is 5 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √55 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 55, 5 and 11 appear an odd number of times, so √55 is irrational and 7.4161984871 is a rounded value.
Where √55 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √55 lies between 7 and 8. 55 is 6 above 49 and 9 below 64, so the root is closer to 7.
- Straight line between 49 and 64: 7.4000 (0.22% low)
- Tangent from 7, i.e. 7 + 6 ÷ 14: 7.4286 (0.17% high)
- Tangent from 8, i.e. 8 − 9 ÷ 16: 7.4375 (0.29% high)
For √55 the tangent at 7 wins, missing by only 0.0124. Tangent estimates shine when the number sits close to a perfect square — here 55 is just 6 above 49.
Finding √55 with the Babylonian method
If a guess is too big, 55 ÷ 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√55) in one step.
Start from the nearest whole number, 7 (7² = 49):
| Step | Guess x | 55 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 7.0000000000 | 7.8571428571 | 7.4285714286 | 1 |
| 2 | 7.4285714286 | 7.4038461538 | 7.4162087912 | 4 |
| 3 | 7.4162087912 | 7.4161881830 | 7.4161984871 | all 10 shown |
The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √55 = 7.4161984871 to every decimal shown.
√55 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √55 the pattern is [7; 2, 2, 2, 14] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √55 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 |
|---|---|---|
| 7/1 | 7.0000000000 | 4.2 × 10⁻¹ |
| 15/2 | 7.5000000000 | 8.4 × 10⁻² |
| 37/5 | 7.4000000000 | 1.6 × 10⁻² |
| 89/12 | 7.4166666667 | 4.7 × 10⁻⁴ |
| 1,283/173 | 7.4161849711 | 1.4 × 10⁻⁵ |
| 2,655/358 | 7.4162011173 | 2.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 55y² = 1. Its smallest solution in positive whole numbers is x = 89, y = 12.
√55 in geometry and everyday measurements
- A square room or garden bed covering 55 square feet measures about 7.42 ft (7 ft 5 in) along each wall.
- 55 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √55 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √55 as its space diagonal.
Square roots near √55 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √52 | 2√13 | 7.2111 | No |
| √53 | √53 | 7.2801 | No |
| √54 | 3√6 | 7.3485 | No |
| √55 | √55 | 7.4162 | No |
| √56 | 2√14 | 7.4833 | No |
| √57 | √57 | 7.5498 | No |
| √58 | √58 | 7.6158 | No |
- The cube root of 55 is about 3.802952.
- Four times the radicand doubles the root: √220 = 2 × √55 ≈ 14.832397.
Frequently asked questions
What is the square root of 55?
The square root of 55 is √55, about 7.4161984871. The negative root, −7.416198, also squares to 55.
Is the square root of 55 rational or irrational?
Irrational. 55 is not a perfect square — it falls between 49 and 64 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √55 be simplified?
No. 55 = 5 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √55 rounded to two decimal places?
√55 ≈ 7.42 to two decimal places (7.4 to one, 7.416 to three). Check: 7.42² = 55.0564, close to 55.