√81 at a glance
- Exact value
- 9
- Decimal (10 places)
- 9
- Rounded
- 9
- Perfect square?
- Yes — 9²
- Rational?
- Rational (whole number)
- Both square roots
- ±9
- Prime factorization
- 3⁴
- Cube root
- 4.326749
How to simplify √81
Write 81 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√81 = 3² = 9
81 has 3 factor pairs (1 × 81, 3 × 27, 9 × 9). The pair with two equal numbers, 9 × 9, is what makes it a perfect square — and why it has an odd number of factors.
Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 81 = 1 + 3 + 5 + 7 + … + 17, the first 9 odd numbers.
Where √81 sits between perfect squares
81 is 9². The perfect squares on either side are 64 (8²) and 100 (10²), so the gaps are 17 below and 19 above — consecutive squares always differ by consecutive odd numbers.
Finding √81 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 81: following the tangent line down to zero simplifies to averaging x with 81 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 10:
| Step | Guess x | 81 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 8.1000000000 | 9.0500000000 | 1 |
| 2 | 9.0500000000 | 8.9502762431 | 9.0001381215 | 3 |
| 3 | 9.0001381215 | 8.9998618806 | 9.0000000011 | 8 |
| 4 | 9.0000000011 | 8.9999999989 | 9.0000000000 | all 10 shown |
The count of correct decimals went 1, 3, 8 and all 10 over 4 steps — roughly doubling each time — until the guess matched √81 = 9.0000000000 to every decimal shown.
√81 in geometry and everyday measurements
- A square room or garden bed covering 81 square feet measures exactly 9 ft (9 ft) along each wall.
- 81 is not a sum of two whole-number squares — no pair of squares adds up to it — so √81 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 8 box, because 1² + 4² + 8² = 81.
Square roots near √81 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √78 | √78 | 8.8318 | No |
| √79 | √79 | 8.8882 | No |
| √80 | 4√5 | 8.9443 | No |
| √81 | 9 | 9.0000 | Yes |
| √82 | √82 | 9.0554 | No |
| √83 | √83 | 9.1104 | No |
| √84 | 2√21 | 9.1652 | No |
- The cube root of 81 is about 4.326749.
- Taking the square root twice gives the fourth root: √9 = 3, so 81 = 3⁴.
- Four times the radicand doubles the root: √324 = 2 × √81 ≈ 18.
Frequently asked questions
What is the square root of 81?
The square root of 81 is 9, because 9 × 9 = 81. Strictly, 81 has two square roots, 9 and −9; the √ symbol means the positive one.
Is the square root of 81 rational or irrational?
Rational. 81 is a perfect square, so its square root is the whole number 9.
Can √81 be simplified?
Yes, all the way to a whole number: √81 = 9.
What is √81 rounded to two decimal places?
√81 is exactly 9, or 9.00 written to two decimal places.