Square Root of 52
The square root of 52 is approximately 7.211103, or exactly 2√13 in simplified radical form. No, 52 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √52 sits
7² = 49 and 8² = 64, and 52 falls between them, so √52 is between 7 and 8. It's closer to 7 (3 above 49, 12 below 64), which matches the decimal 7.211103.
| k | k² | Compared to 52 |
|---|---|---|
| 5 | 25 | 27 below |
| 6 | 36 | 16 below |
| 7 | 49 | 3 below |
| 8 | 64 | 12 above |
| 9 | 81 | 29 above |
| 10 | 100 | 48 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 52 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 52 ÷ x | New guess (x + 52÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 7.428571 | 7.214286 |
| 2 | 7.214286 | 7.207921 | 7.211103 |
| 3 | 7.211103 | 7.211102 | 7.211103 |
| 4 | 7.211103 | 7.211103 | 7.211103 |
After 4 passes the guess is 7.211103, already matching √52 = 7.211103 to six decimal places.
Exact form from the prime factors
52 = 2^2 × 13. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √52 = 2√13. That's the exact value; the decimal 7.211103 is a rounding of it.
Frequently Asked Questions
What is the square root of 52?
The square root of 52 is approximately 7.211103, or exactly 2√13 in simplified radical form.
Is 52 a perfect square?
No, 52 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √52 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 52?
√51 is about 7.1414, and √53 is about 7.2801. 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 √52 squared?
Exactly 52 -- squaring undoes the square root. Squaring the rounded decimal 7.211103 gives 52, which is why the radical form 2√13 is the exact answer and the decimal is an approximation.
What is the negative square root of 52?
−7.211103. Every positive number has two square roots, one positive and one negative, because (−7.211103)² is also 52. The radical sign √ by convention means the positive one.
More Number Theory
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