Square Root of 40
The square root of 40 is approximately 6.324555, or exactly 2√10 in simplified radical form. No, 40 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √40 sits
6² = 36 and 7² = 49, and 40 falls between them, so √40 is between 6 and 7. It's closer to 6 (4 above 36, 9 below 49), which matches the decimal 6.324555.
| k | k² | Compared to 40 |
|---|---|---|
| 4 | 16 | 24 below |
| 5 | 25 | 15 below |
| 6 | 36 | 4 below |
| 7 | 49 | 9 above |
| 8 | 64 | 24 above |
| 9 | 81 | 41 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 40 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 40 ÷ x | New guess (x + 40÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 6.666667 | 6.333333 |
| 2 | 6.333333 | 6.315789 | 6.324561 |
| 3 | 6.324561 | 6.324549 | 6.324555 |
| 4 | 6.324555 | 6.324555 | 6.324555 |
After 4 passes the guess is 6.324555, already matching √40 = 6.324555 to six decimal places.
Exact form from the prime factors
40 = 2^3 × 5. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √40 = 2√10. That's the exact value; the decimal 6.324555 is a rounding of it.
Frequently Asked Questions
What is the square root of 40?
The square root of 40 is approximately 6.324555, or exactly 2√10 in simplified radical form.
Is 40 a perfect square?
No, 40 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √40 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 40?
√39 is about 6.245, and √41 is about 6.4031. 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 √40 squared?
Exactly 40 -- squaring undoes the square root. Squaring the rounded decimal 6.324555 gives 40, which is why the radical form 2√10 is the exact answer and the decimal is an approximation.
What is the negative square root of 40?
−6.324555. Every positive number has two square roots, one positive and one negative, because (−6.324555)² is also 40. The radical sign √ by convention means the positive one.
More Number Theory
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