Square root of 84
√84 ≈ 9.1652. The square root of 84 is about 9.1652, which simplifies to 2√21. 84 is not a perfect square, so √84 is irrational: its decimals never end or repeat.
| Value | 9.16515139 |
|---|---|
| Rounded | 9.2 (1 dp) · 9.17 (2 dp) · 9.165 (3 dp) |
| Simplest radical form | 2√21 |
| Perfect square? | No — between 9² = 81 and 10² = 100 |
| 84 squared | 7,056 |
Estimating √84 in your head
Start from the nearest perfect square, 9² = 81. 84 is 3 more than that. Divide the difference by twice the root, 18, and add it:
√84 ≈ 9 + 3 ÷ 18 ≈ 9.167
The true value is 9.1652, so the one-step estimate is off by about 0.002. The estimate always comes out slightly high, and it is closest when 84 is near a perfect square. The method is explained in estimating square roots in your head.
Simplifying √84
Find the largest perfect square that divides 84: it is 4, since 84 = 4 × 21. Take its root outside: √84 = √4 × √21 = 2√21.
Frequently asked questions
What is the square root of 84?
The square root of 84 is about 9.1652, which simplifies to 2√21. 84 is not a perfect square, so √84 is irrational: its decimals never end or repeat.
Is 84 a perfect square?
No. It lies between 9² = 81 and 10² = 100.
What is √84 in simplest radical form?
√84 = √(4 × 21) = 2√21.
Practise estimating square rootsNearby: √83 · √85 · all square roots to 150