Square Root of 55
The square root of 55 is approximately 7.416198, or exactly √55 in simplified radical form. No, 55 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √55 sits
7² = 49 and 8² = 64, and 55 falls between them, so √55 is between 7 and 8. It's closer to 7 (6 above 49, 9 below 64), which matches the decimal 7.416198.
| k | k² | Compared to 55 |
|---|---|---|
| 5 | 25 | 30 below |
| 6 | 36 | 19 below |
| 7 | 49 | 6 below |
| 8 | 64 | 9 above |
| 9 | 81 | 26 above |
| 10 | 100 | 45 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 55 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 55 ÷ x | New guess (x + 55÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 7.857143 | 7.428571 |
| 2 | 7.428571 | 7.403846 | 7.416209 |
| 3 | 7.416209 | 7.416188 | 7.416198 |
| 4 | 7.416198 | 7.416198 | 7.416198 |
After 4 passes the guess is 7.416198, already matching √55 = 7.416198 to six decimal places.
Exact form from the prime factors
55 = 5 × 11. No prime appears twice, so nothing can be pulled out of the root -- √55 is already in simplest form.
Frequently Asked Questions
What is the square root of 55?
The square root of 55 is approximately 7.416198, or exactly √55 in simplified radical form.
Is 55 a perfect square?
No, 55 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √55 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 55?
√54 is about 7.3485, and √56 is about 7.4833. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √55 squared?
Exactly 55 -- squaring undoes the square root. Squaring the rounded decimal 7.416198 gives 55, which is why the radical form √55 is the exact answer and the decimal is an approximation.
What is the negative square root of 55?
−7.416198. Every positive number has two square roots, one positive and one negative, because (−7.416198)² is also 55. The radical sign √ by convention means the positive one.
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