Square Root of 59
The square root of 59 is approximately 7.681146, or exactly √59 in simplified radical form. No, 59 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √59 sits
7² = 49 and 8² = 64, and 59 falls between them, so √59 is between 7 and 8. It's closer to 8 (10 above 49, 5 below 64), which matches the decimal 7.681146.
| k | k² | Compared to 59 |
|---|---|---|
| 5 | 25 | 34 below |
| 6 | 36 | 23 below |
| 7 | 49 | 10 below |
| 8 | 64 | 5 above |
| 9 | 81 | 22 above |
| 10 | 100 | 41 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 59 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 59 ÷ x | New guess (x + 59÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 8.428571 | 7.714286 |
| 2 | 7.714286 | 7.648148 | 7.681217 |
| 3 | 7.681217 | 7.681075 | 7.681146 |
| 4 | 7.681146 | 7.681146 | 7.681146 |
After 4 passes the guess is 7.681146, already matching √59 = 7.681146 to six decimal places.
Exact form from the prime factors
59 = 59. No prime appears twice, so nothing can be pulled out of the root -- √59 is already in simplest form.
Frequently Asked Questions
What is the square root of 59?
The square root of 59 is approximately 7.681146, or exactly √59 in simplified radical form.
Is 59 a perfect square?
No, 59 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √59 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 59?
√58 is about 7.6158, and √60 is about 7.746. 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 √59 squared?
Exactly 59 -- squaring undoes the square root. Squaring the rounded decimal 7.681146 gives 59, which is why the radical form √59 is the exact answer and the decimal is an approximation.
What is the negative square root of 59?
−7.681146. Every positive number has two square roots, one positive and one negative, because (−7.681146)² is also 59. The radical sign √ by convention means the positive one.
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