Square Root of 39
The square root of 39 is approximately 6.244998, or exactly √39 in simplified radical form. No, 39 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √39 sits
6² = 36 and 7² = 49, and 39 falls between them, so √39 is between 6 and 7. It's closer to 6 (3 above 36, 10 below 49), which matches the decimal 6.244998.
| k | k² | Compared to 39 |
|---|---|---|
| 4 | 16 | 23 below |
| 5 | 25 | 14 below |
| 6 | 36 | 3 below |
| 7 | 49 | 10 above |
| 8 | 64 | 25 above |
| 9 | 81 | 42 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 39 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 39 ÷ x | New guess (x + 39÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 6.5 | 6.25 |
| 2 | 6.25 | 6.24 | 6.245 |
| 3 | 6.245 | 6.244996 | 6.244998 |
| 4 | 6.244998 | 6.244998 | 6.244998 |
After 4 passes the guess is 6.244998, already matching √39 = 6.244998 to six decimal places.
Exact form from the prime factors
39 = 3 × 13. No prime appears twice, so nothing can be pulled out of the root -- √39 is already in simplest form.
Frequently Asked Questions
What is the square root of 39?
The square root of 39 is approximately 6.244998, or exactly √39 in simplified radical form.
Is 39 a perfect square?
No, 39 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √39 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 39?
√38 is about 6.1644, and √40 is about 6.3246. 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 √39 squared?
Exactly 39 -- squaring undoes the square root. Squaring the rounded decimal 6.244998 gives 39, which is why the radical form √39 is the exact answer and the decimal is an approximation.
What is the negative square root of 39?
−6.244998. Every positive number has two square roots, one positive and one negative, because (−6.244998)² is also 39. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 39
- GCF Calculator
- LCM Calculator