Square Root of 31
The square root of 31 is approximately 5.567764, or exactly √31 in simplified radical form. No, 31 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √31 sits
5² = 25 and 6² = 36, and 31 falls between them, so √31 is between 5 and 6. It's closer to 6 (6 above 25, 5 below 36), which matches the decimal 5.567764.
| k | k² | Compared to 31 |
|---|---|---|
| 3 | 9 | 22 below |
| 4 | 16 | 15 below |
| 5 | 25 | 6 below |
| 6 | 36 | 5 above |
| 7 | 49 | 18 above |
| 8 | 64 | 33 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 31 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 31 ÷ x | New guess (x + 31÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 6.2 | 5.6 |
| 2 | 5.6 | 5.535714 | 5.567857 |
| 3 | 5.567857 | 5.567672 | 5.567764 |
| 4 | 5.567764 | 5.567764 | 5.567764 |
After 4 passes the guess is 5.567764, already matching √31 = 5.567764 to six decimal places.
Exact form from the prime factors
31 = 31. No prime appears twice, so nothing can be pulled out of the root -- √31 is already in simplest form.
Frequently Asked Questions
What is the square root of 31?
The square root of 31 is approximately 5.567764, or exactly √31 in simplified radical form.
Is 31 a perfect square?
No, 31 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √31 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 31?
√30 is about 5.4772, and √32 is about 5.6569. 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 √31 squared?
Exactly 31 -- squaring undoes the square root. Squaring the rounded decimal 5.567764 gives 31, which is why the radical form √31 is the exact answer and the decimal is an approximation.
What is the negative square root of 31?
−5.567764. Every positive number has two square roots, one positive and one negative, because (−5.567764)² is also 31. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 31
- GCF Calculator
- LCM Calculator