Square Root of 61
The square root of 61 is approximately 7.81025, or exactly √61 in simplified radical form. No, 61 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √61 sits
7² = 49 and 8² = 64, and 61 falls between them, so √61 is between 7 and 8. It's closer to 8 (12 above 49, 3 below 64), which matches the decimal 7.81025.
| k | k² | Compared to 61 |
|---|---|---|
| 5 | 25 | 36 below |
| 6 | 36 | 25 below |
| 7 | 49 | 12 below |
| 8 | 64 | 3 above |
| 9 | 81 | 20 above |
| 10 | 100 | 39 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 61 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 61 ÷ x | New guess (x + 61÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 8.714286 | 7.857143 |
| 2 | 7.857143 | 7.763636 | 7.81039 |
| 3 | 7.81039 | 7.81011 | 7.81025 |
| 4 | 7.81025 | 7.81025 | 7.81025 |
After 4 passes the guess is 7.81025, already matching √61 = 7.81025 to six decimal places.
Exact form from the prime factors
61 = 61. No prime appears twice, so nothing can be pulled out of the root -- √61 is already in simplest form.
Frequently Asked Questions
What is the square root of 61?
The square root of 61 is approximately 7.81025, or exactly √61 in simplified radical form.
Is 61 a perfect square?
No, 61 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √61 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 61?
√60 is about 7.746, and √62 is about 7.874. 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 √61 squared?
Exactly 61 -- squaring undoes the square root. Squaring the rounded decimal 7.81025 gives 61, which is why the radical form √61 is the exact answer and the decimal is an approximation.
What is the negative square root of 61?
−7.81025. Every positive number has two square roots, one positive and one negative, because (−7.81025)² is also 61. The radical sign √ by convention means the positive one.
More Number Theory
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