Square Root of 63
The square root of 63 is approximately 7.937254, or exactly 3√7 in simplified radical form. No, 63 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √63 sits
7² = 49 and 8² = 64, and 63 falls between them, so √63 is between 7 and 8. It's closer to 8 (14 above 49, 1 below 64), which matches the decimal 7.937254.
| k | k² | Compared to 63 |
|---|---|---|
| 5 | 25 | 38 below |
| 6 | 36 | 27 below |
| 7 | 49 | 14 below |
| 8 | 64 | 1 above |
| 9 | 81 | 18 above |
| 10 | 100 | 37 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 63 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 63 ÷ x | New guess (x + 63÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 9 | 8 |
| 2 | 8 | 7.875 | 7.9375 |
| 3 | 7.9375 | 7.937008 | 7.937254 |
| 4 | 7.937254 | 7.937254 | 7.937254 |
After 4 passes the guess is 7.937254, already matching √63 = 7.937254 to six decimal places.
Exact form from the prime factors
63 = 3^2 × 7. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √63 = 3√7. That's the exact value; the decimal 7.937254 is a rounding of it.
Frequently Asked Questions
What is the square root of 63?
The square root of 63 is approximately 7.937254, or exactly 3√7 in simplified radical form.
Is 63 a perfect square?
No, 63 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √63 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 63?
√62 is about 7.874, and √64 is exactly 8. 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 √63 squared?
Exactly 63 -- squaring undoes the square root. Squaring the rounded decimal 7.937254 gives 63, which is why the radical form 3√7 is the exact answer and the decimal is an approximation.
What is the negative square root of 63?
−7.937254. Every positive number has two square roots, one positive and one negative, because (−7.937254)² is also 63. The radical sign √ by convention means the positive one.
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