Square Root of 94
The square root of 94 is approximately 9.69536, or exactly √94 in simplified radical form. No, 94 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √94 sits
9² = 81 and 10² = 100, and 94 falls between them, so √94 is between 9 and 10. It's closer to 10 (13 above 81, 6 below 100), which matches the decimal 9.69536.
| k | k² | Compared to 94 |
|---|---|---|
| 7 | 49 | 45 below |
| 8 | 64 | 30 below |
| 9 | 81 | 13 below |
| 10 | 100 | 6 above |
| 11 | 121 | 27 above |
| 12 | 144 | 50 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 94 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 94 ÷ x | New guess (x + 94÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 10.444444 | 9.722222 |
| 2 | 9.722222 | 9.668571 | 9.695397 |
| 3 | 9.695397 | 9.695323 | 9.69536 |
| 4 | 9.69536 | 9.69536 | 9.69536 |
After 4 passes the guess is 9.69536, already matching √94 = 9.69536 to six decimal places.
Exact form from the prime factors
94 = 2 × 47. No prime appears twice, so nothing can be pulled out of the root -- √94 is already in simplest form.
Frequently Asked Questions
What is the square root of 94?
The square root of 94 is approximately 9.69536, or exactly √94 in simplified radical form.
Is 94 a perfect square?
No, 94 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √94 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 94?
√93 is about 9.6437, and √95 is about 9.7468. 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 √94 squared?
Exactly 94 -- squaring undoes the square root. Squaring the rounded decimal 9.69536 gives 94, which is why the radical form √94 is the exact answer and the decimal is an approximation.
What is the negative square root of 94?
−9.69536. Every positive number has two square roots, one positive and one negative, because (−9.69536)² is also 94. The radical sign √ by convention means the positive one.
More Number Theory
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