Square Root of 37
The square root of 37 is approximately 6.082763, or exactly √37 in simplified radical form. No, 37 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √37 sits
6² = 36 and 7² = 49, and 37 falls between them, so √37 is between 6 and 7. It's closer to 6 (1 above 36, 12 below 49), which matches the decimal 6.082763.
| k | k² | Compared to 37 |
|---|---|---|
| 4 | 16 | 21 below |
| 5 | 25 | 12 below |
| 6 | 36 | 1 below |
| 7 | 49 | 12 above |
| 8 | 64 | 27 above |
| 9 | 81 | 44 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 37 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 37 ÷ x | New guess (x + 37÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 6.166667 | 6.083333 |
| 2 | 6.083333 | 6.082192 | 6.082763 |
| 3 | 6.082763 | 6.082763 | 6.082763 |
| 4 | 6.082763 | 6.082763 | 6.082763 |
After 4 passes the guess is 6.082763, already matching √37 = 6.082763 to six decimal places.
Exact form from the prime factors
37 = 37. No prime appears twice, so nothing can be pulled out of the root -- √37 is already in simplest form.
Frequently Asked Questions
What is the square root of 37?
The square root of 37 is approximately 6.082763, or exactly √37 in simplified radical form.
Is 37 a perfect square?
No, 37 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √37 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 37?
√36 is exactly 6, and √38 is about 6.1644. 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 √37 squared?
Exactly 37 -- squaring undoes the square root. Squaring the rounded decimal 6.082763 gives 37, which is why the radical form √37 is the exact answer and the decimal is an approximation.
What is the negative square root of 37?
−6.082763. Every positive number has two square roots, one positive and one negative, because (−6.082763)² is also 37. The radical sign √ by convention means the positive one.
More Number Theory
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