Square Root of 58
The square root of 58 is approximately 7.615773, or exactly √58 in simplified radical form. No, 58 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √58 sits
7² = 49 and 8² = 64, and 58 falls between them, so √58 is between 7 and 8. It's closer to 8 (9 above 49, 6 below 64), which matches the decimal 7.615773.
| k | k² | Compared to 58 |
|---|---|---|
| 5 | 25 | 33 below |
| 6 | 36 | 22 below |
| 7 | 49 | 9 below |
| 8 | 64 | 6 above |
| 9 | 81 | 23 above |
| 10 | 100 | 42 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 58 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 58 ÷ x | New guess (x + 58÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 8.285714 | 7.642857 |
| 2 | 7.642857 | 7.588785 | 7.615821 |
| 3 | 7.615821 | 7.615725 | 7.615773 |
| 4 | 7.615773 | 7.615773 | 7.615773 |
After 4 passes the guess is 7.615773, already matching √58 = 7.615773 to six decimal places.
Exact form from the prime factors
58 = 2 × 29. No prime appears twice, so nothing can be pulled out of the root -- √58 is already in simplest form.
Frequently Asked Questions
What is the square root of 58?
The square root of 58 is approximately 7.615773, or exactly √58 in simplified radical form.
Is 58 a perfect square?
No, 58 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √58 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 58?
√57 is about 7.5498, and √59 is about 7.6811. 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 √58 squared?
Exactly 58 -- squaring undoes the square root. Squaring the rounded decimal 7.615773 gives 58, which is why the radical form √58 is the exact answer and the decimal is an approximation.
What is the negative square root of 58?
−7.615773. Every positive number has two square roots, one positive and one negative, because (−7.615773)² is also 58. The radical sign √ by convention means the positive one.
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