Square Root of 53
The square root of 53 is approximately 7.28011, or exactly √53 in simplified radical form. No, 53 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √53 sits
7² = 49 and 8² = 64, and 53 falls between them, so √53 is between 7 and 8. It's closer to 7 (4 above 49, 11 below 64), which matches the decimal 7.28011.
| k | k² | Compared to 53 |
|---|---|---|
| 5 | 25 | 28 below |
| 6 | 36 | 17 below |
| 7 | 49 | 4 below |
| 8 | 64 | 11 above |
| 9 | 81 | 28 above |
| 10 | 100 | 47 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 53 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 53 ÷ x | New guess (x + 53÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 7.571429 | 7.285714 |
| 2 | 7.285714 | 7.27451 | 7.280112 |
| 3 | 7.280112 | 7.280108 | 7.28011 |
| 4 | 7.28011 | 7.28011 | 7.28011 |
After 4 passes the guess is 7.28011, already matching √53 = 7.28011 to six decimal places.
Exact form from the prime factors
53 = 53. No prime appears twice, so nothing can be pulled out of the root -- √53 is already in simplest form.
Frequently Asked Questions
What is the square root of 53?
The square root of 53 is approximately 7.28011, or exactly √53 in simplified radical form.
Is 53 a perfect square?
No, 53 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √53 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 53?
√52 is about 7.2111, and √54 is about 7.3485. 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 √53 squared?
Exactly 53 -- squaring undoes the square root. Squaring the rounded decimal 7.28011 gives 53, which is why the radical form √53 is the exact answer and the decimal is an approximation.
What is the negative square root of 53?
−7.28011. Every positive number has two square roots, one positive and one negative, because (−7.28011)² is also 53. The radical sign √ by convention means the positive one.
More Number Theory
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