Square Root of 90
The square root of 90 is approximately 9.486833, or exactly 3√10 in simplified radical form. No, 90 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √90 sits
9² = 81 and 10² = 100, and 90 falls between them, so √90 is between 9 and 10. It's closer to 9 (9 above 81, 10 below 100), which matches the decimal 9.486833.
| k | k² | Compared to 90 |
|---|---|---|
| 7 | 49 | 41 below |
| 8 | 64 | 26 below |
| 9 | 81 | 9 below |
| 10 | 100 | 10 above |
| 11 | 121 | 31 above |
| 12 | 144 | 54 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 90 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 90 ÷ x | New guess (x + 90÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 10 | 9.5 |
| 2 | 9.5 | 9.473684 | 9.486842 |
| 3 | 9.486842 | 9.486824 | 9.486833 |
| 4 | 9.486833 | 9.486833 | 9.486833 |
After 4 passes the guess is 9.486833, already matching √90 = 9.486833 to six decimal places.
Exact form from the prime factors
90 = 2 × 3^2 × 5. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √90 = 3√10. That's the exact value; the decimal 9.486833 is a rounding of it.
Frequently Asked Questions
What is the square root of 90?
The square root of 90 is approximately 9.486833, or exactly 3√10 in simplified radical form.
Is 90 a perfect square?
No, 90 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √90 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 90?
√89 is about 9.434, and √91 is about 9.5394. 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 √90 squared?
Exactly 90 -- squaring undoes the square root. Squaring the rounded decimal 9.486833 gives 90, which is why the radical form 3√10 is the exact answer and the decimal is an approximation.
What is the negative square root of 90?
−9.486833. Every positive number has two square roots, one positive and one negative, because (−9.486833)² is also 90. The radical sign √ by convention means the positive one.
More Number Theory
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