Square Root of 84
The square root of 84 is approximately 9.165151, or exactly 2√21 in simplified radical form. No, 84 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √84 sits
9² = 81 and 10² = 100, and 84 falls between them, so √84 is between 9 and 10. It's closer to 9 (3 above 81, 16 below 100), which matches the decimal 9.165151.
| k | k² | Compared to 84 |
|---|---|---|
| 7 | 49 | 35 below |
| 8 | 64 | 20 below |
| 9 | 81 | 3 below |
| 10 | 100 | 16 above |
| 11 | 121 | 37 above |
| 12 | 144 | 60 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 9), divide 84 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 84 ÷ x | New guess (x + 84÷x) ÷ 2 |
|---|---|---|---|
| 1 | 9 | 9.333333 | 9.166667 |
| 2 | 9.166667 | 9.163636 | 9.165152 |
| 3 | 9.165152 | 9.165151 | 9.165151 |
| 4 | 9.165151 | 9.165151 | 9.165151 |
After 4 passes the guess is 9.165151, already matching √84 = 9.165151 to six decimal places.
Exact form from the prime factors
84 = 2^2 × 3 × 7. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √84 = 2√21. That's the exact value; the decimal 9.165151 is a rounding of it.
Frequently Asked Questions
What is the square root of 84?
The square root of 84 is approximately 9.165151, or exactly 2√21 in simplified radical form.
Is 84 a perfect square?
No, 84 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √84 between?
Between 9 and 10, because 9² = 81 and 10² = 100.
What are the square roots of the numbers right next to 84?
√83 is about 9.1104, and √85 is about 9.2195. 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 √84 squared?
Exactly 84 -- squaring undoes the square root. Squaring the rounded decimal 9.165151 gives 84, which is why the radical form 2√21 is the exact answer and the decimal is an approximation.
What is the negative square root of 84?
−9.165151. Every positive number has two square roots, one positive and one negative, because (−9.165151)² is also 84. The radical sign √ by convention means the positive one.
More Number Theory
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