√9 at a glance
- Exact value
- 3
- Decimal (10 places)
- 3
- Rounded
- 3
- Perfect square?
- Yes — 3²
- Rational?
- Rational (whole number)
- Both square roots
- ±3
- Prime factorization
- 3²
- Cube root
- 2.080084
How to simplify √9
Write 9 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√9 = 3 = 3
9 has 2 factor pairs (1 × 9, 3 × 3). The pair with two equal numbers, 3 × 3, 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 9 = 1 + 3 + 5, the first 3 odd numbers.
Where √9 sits between perfect squares
9 is 3². The perfect squares on either side are 4 (2²) and 16 (4²), so the gaps are 5 below and 7 above — consecutive squares always differ by consecutive odd numbers.
Finding √9 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 9: following the tangent line down to zero simplifies to averaging x with 9 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 4:
| Step | Guess x | 9 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 4.0000000000 | 2.2500000000 | 3.1250000000 | 0 |
| 2 | 3.1250000000 | 2.8800000000 | 3.0025000000 | 2 |
| 3 | 3.0025000000 | 2.9975020816 | 3.0000010408 | 5 |
| 4 | 3.0000010408 | 2.9999989592 | 3.0000000000 | all 10 shown |
The count of correct decimals went 0, 2, 5 and all 10 over 4 steps — roughly doubling each time — until the guess matched √9 = 3.0000000000 to every decimal shown.
√9 in geometry and everyday measurements
- A square tile with an area of 9 square inches has sides exactly 3 in long.
- 9 is not a sum of two whole-number squares — no pair of squares adds up to it — so √9 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 2 box, because 1² + 2² + 2² = 9.
Square roots near √9 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √6 | √6 | 2.4495 | No |
| √7 | √7 | 2.6458 | No |
| √8 | 2√2 | 2.8284 | No |
| √9 | 3 | 3.0000 | Yes |
| √10 | √10 | 3.1623 | No |
| √11 | √11 | 3.3166 | No |
| √12 | 2√3 | 3.4641 | No |
- The cube root of 9 is about 2.080084.
- Taking the square root twice gives the fourth root: √3 ≈ 1.732051.
- Multiplying the radicand by 100 multiplies the root by 10: √900 = 10 × √9 ≈ 30.
Frequently asked questions
What is the square root of 9?
The square root of 9 is 3, because 3 × 3 = 9. Strictly, 9 has two square roots, 3 and −3; the √ symbol means the positive one.
Is the square root of 9 rational or irrational?
Rational. 9 is a perfect square, so its square root is the whole number 3.
Can √9 be simplified?
Yes, all the way to a whole number: √9 = 3.
What is √9 rounded to two decimal places?
√9 is exactly 3, or 3.00 written to two decimal places.