Square Root of 38
The square root of 38 is approximately 6.164414, or exactly √38 in simplified radical form. No, 38 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √38 sits
6² = 36 and 7² = 49, and 38 falls between them, so √38 is between 6 and 7. It's closer to 6 (2 above 36, 11 below 49), which matches the decimal 6.164414.
| k | k² | Compared to 38 |
|---|---|---|
| 4 | 16 | 22 below |
| 5 | 25 | 13 below |
| 6 | 36 | 2 below |
| 7 | 49 | 11 above |
| 8 | 64 | 26 above |
| 9 | 81 | 43 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 38 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 38 ÷ x | New guess (x + 38÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 6.333333 | 6.166667 |
| 2 | 6.166667 | 6.162162 | 6.164414 |
| 3 | 6.164414 | 6.164414 | 6.164414 |
| 4 | 6.164414 | 6.164414 | 6.164414 |
After 4 passes the guess is 6.164414, already matching √38 = 6.164414 to six decimal places.
Exact form from the prime factors
38 = 2 × 19. No prime appears twice, so nothing can be pulled out of the root -- √38 is already in simplest form.
Frequently Asked Questions
What is the square root of 38?
The square root of 38 is approximately 6.164414, or exactly √38 in simplified radical form.
Is 38 a perfect square?
No, 38 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √38 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 38?
√37 is about 6.0828, and √39 is about 6.245. 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 √38 squared?
Exactly 38 -- squaring undoes the square root. Squaring the rounded decimal 6.164414 gives 38, which is why the radical form √38 is the exact answer and the decimal is an approximation.
What is the negative square root of 38?
−6.164414. Every positive number has two square roots, one positive and one negative, because (−6.164414)² is also 38. The radical sign √ by convention means the positive one.
More Number Theory
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