Square Root Calculator › Square root of 84
Square Root of 84
√84 ≈ 9.165151
84 is not a perfect square, so √84 is an irrational number that is approximately 9.165151. In simplified radical form, √84 = 2√21.
√84 lies between √81 = 9 and √100 = 10.
| Number | 84 |
| Square root (decimal) | 9.165151 |
| Simplified radical | 2√21 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 84
84 = 2^2 × 3 × 7. This is what determines the simplified radical: for √84 you pull out one copy of any prime that appears twice.
Here the paired factor 2 comes out from under the root, leaving √84 = 2√21.
Estimating √84 by hand
√84 falls between √81 = 9 and √100 = 10, so it starts at 9.something. The Babylonian method refines that guess by averaging it with 84 ÷ the guess:
| Step | Estimate |
| Start | 9 |
| Step 1 | 9.166667 |
| Step 2 | 9.165152 |
| Step 3 | 9.165151 |
After 3 steps the estimate is 9.165151, matching √84 ≈ 9.165151.
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Frequently asked questions
Is 84 a perfect square?
No, 84 is not a perfect square, so its square root is irrational and approximately 9.165151.
What is the square root of 84 in simplest radical form?
√84 = 2√21 in simplest radical form.