Square Root of 57
The square root of 57 is approximately 7.549834, or exactly √57 in simplified radical form. No, 57 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √57 sits
7² = 49 and 8² = 64, and 57 falls between them, so √57 is between 7 and 8. It's closer to 8 (8 above 49, 7 below 64), which matches the decimal 7.549834.
| k | k² | Compared to 57 |
|---|---|---|
| 5 | 25 | 32 below |
| 6 | 36 | 21 below |
| 7 | 49 | 8 below |
| 8 | 64 | 7 above |
| 9 | 81 | 24 above |
| 10 | 100 | 43 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 57 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 57 ÷ x | New guess (x + 57÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 8.142857 | 7.571429 |
| 2 | 7.571429 | 7.528302 | 7.549865 |
| 3 | 7.549865 | 7.549804 | 7.549834 |
| 4 | 7.549834 | 7.549834 | 7.549834 |
After 4 passes the guess is 7.549834, already matching √57 = 7.549834 to six decimal places.
Exact form from the prime factors
57 = 3 × 19. No prime appears twice, so nothing can be pulled out of the root -- √57 is already in simplest form.
Frequently Asked Questions
What is the square root of 57?
The square root of 57 is approximately 7.549834, or exactly √57 in simplified radical form.
Is 57 a perfect square?
No, 57 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √57 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 57?
√56 is about 7.4833, and √58 is about 7.6158. 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 √57 squared?
Exactly 57 -- squaring undoes the square root. Squaring the rounded decimal 7.549834 gives 57, which is why the radical form √57 is the exact answer and the decimal is an approximation.
What is the negative square root of 57?
−7.549834. Every positive number has two square roots, one positive and one negative, because (−7.549834)² is also 57. The radical sign √ by convention means the positive one.
More Number Theory
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