Square Root of 60
The square root of 60 is approximately 7.745967, or exactly 2√15 in simplified radical form. No, 60 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √60 sits
7² = 49 and 8² = 64, and 60 falls between them, so √60 is between 7 and 8. It's closer to 8 (11 above 49, 4 below 64), which matches the decimal 7.745967.
| k | k² | Compared to 60 |
|---|---|---|
| 5 | 25 | 35 below |
| 6 | 36 | 24 below |
| 7 | 49 | 11 below |
| 8 | 64 | 4 above |
| 9 | 81 | 21 above |
| 10 | 100 | 40 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 60 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 60 ÷ x | New guess (x + 60÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 8.571429 | 7.785714 |
| 2 | 7.785714 | 7.706422 | 7.746068 |
| 3 | 7.746068 | 7.745865 | 7.745967 |
| 4 | 7.745967 | 7.745967 | 7.745967 |
After 4 passes the guess is 7.745967, already matching √60 = 7.745967 to six decimal places.
Exact form from the prime factors
60 = 2^2 × 3 × 5. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √60 = 2√15. That's the exact value; the decimal 7.745967 is a rounding of it.
Frequently Asked Questions
What is the square root of 60?
The square root of 60 is approximately 7.745967, or exactly 2√15 in simplified radical form.
Is 60 a perfect square?
No, 60 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √60 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 60?
√59 is about 7.6811, and √61 is about 7.8102. 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 √60 squared?
Exactly 60 -- squaring undoes the square root. Squaring the rounded decimal 7.745967 gives 60, which is why the radical form 2√15 is the exact answer and the decimal is an approximation.
What is the negative square root of 60?
−7.745967. Every positive number has two square roots, one positive and one negative, because (−7.745967)² is also 60. The radical sign √ by convention means the positive one.
More Number Theory
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