Square Root of 93
The square root of 93 is approximately 9.643651, or exactly √93 in simplified radical form. No, 93 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √93 sits
9² = 81 and 10² = 100, and 93 falls between them, so √93 is between 9 and 10. It's closer to 10 (12 above 81, 7 below 100), which matches the decimal 9.643651.
| k | k² | Compared to 93 |
|---|---|---|
| 7 | 49 | 44 below |
| 8 | 64 | 29 below |
| 9 | 81 | 12 below |
| 10 | 100 | 7 above |
| 11 | 121 | 28 above |
| 12 | 144 | 51 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 93 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 93 ÷ x | New guess (x + 93÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 10.333333 | 9.666667 |
| 2 | 9.666667 | 9.62069 | 9.643678 |
| 3 | 9.643678 | 9.643623 | 9.643651 |
| 4 | 9.643651 | 9.643651 | 9.643651 |
After 4 passes the guess is 9.643651, already matching √93 = 9.643651 to six decimal places.
Exact form from the prime factors
93 = 3 × 31. No prime appears twice, so nothing can be pulled out of the root -- √93 is already in simplest form.
Frequently Asked Questions
What is the square root of 93?
The square root of 93 is approximately 9.643651, or exactly √93 in simplified radical form.
Is 93 a perfect square?
No, 93 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √93 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 93?
√92 is about 9.5917, and √94 is about 9.6954. 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 √93 squared?
Exactly 93 -- squaring undoes the square root. Squaring the rounded decimal 9.643651 gives 93, which is why the radical form √93 is the exact answer and the decimal is an approximation.
What is the negative square root of 93?
−9.643651. Every positive number has two square roots, one positive and one negative, because (−9.643651)² is also 93. The radical sign √ by convention means the positive one.
More Number Theory
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