Square Root of 87
The square root of 87 is approximately 9.327379, or exactly √87 in simplified radical form. No, 87 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √87 sits
9² = 81 and 10² = 100, and 87 falls between them, so √87 is between 9 and 10. It's closer to 9 (6 above 81, 13 below 100), which matches the decimal 9.327379.
| k | k² | Compared to 87 |
|---|---|---|
| 7 | 49 | 38 below |
| 8 | 64 | 23 below |
| 9 | 81 | 6 below |
| 10 | 100 | 13 above |
| 11 | 121 | 34 above |
| 12 | 144 | 57 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 87 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 87 ÷ x | New guess (x + 87÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 9.666667 | 9.333333 |
| 2 | 9.333333 | 9.321429 | 9.327381 |
| 3 | 9.327381 | 9.327377 | 9.327379 |
| 4 | 9.327379 | 9.327379 | 9.327379 |
After 4 passes the guess is 9.327379, already matching √87 = 9.327379 to six decimal places.
Exact form from the prime factors
87 = 3 × 29. No prime appears twice, so nothing can be pulled out of the root -- √87 is already in simplest form.
Frequently Asked Questions
What is the square root of 87?
The square root of 87 is approximately 9.327379, or exactly √87 in simplified radical form.
Is 87 a perfect square?
No, 87 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √87 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 87?
√86 is about 9.2736, and √88 is about 9.3808. 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 √87 squared?
Exactly 87 -- squaring undoes the square root. Squaring the rounded decimal 9.327379 gives 87, which is why the radical form √87 is the exact answer and the decimal is an approximation.
What is the negative square root of 87?
−9.327379. Every positive number has two square roots, one positive and one negative, because (−9.327379)² is also 87. The radical sign √ by convention means the positive one.
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