√36 at a glance
- Exact value
- 6
- Decimal (10 places)
- 6
- Rounded
- 6
- Perfect square?
- Yes — 6²
- Rational?
- Rational (whole number)
- Both square roots
- ±6
- Prime factorization
- 2² × 3²
- Cube root
- 3.301927
How to simplify √36
Write 36 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√36 = 2 × 3 = 6
36 has 5 factor pairs (1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6). The pair with two equal numbers, 6 × 6, 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 36 = 1 + 3 + 5 + 7 + … + 11, the first 6 odd numbers.
Where √36 sits between perfect squares
36 is 6². The perfect squares on either side are 25 (5²) and 49 (7²), so the gaps are 11 below and 13 above — consecutive squares always differ by consecutive odd numbers.
Finding √36 with the Babylonian method
Picture a rectangle with an area of 36 and one side x; the other side must be 36 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √36.
To show the method at work on a perfect square, start one above the answer, at 7:
| Step | Guess x | 36 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 7.0000000000 | 5.1428571429 | 6.0714285714 | 1 |
| 2 | 6.0714285714 | 5.9294117647 | 6.0004201681 | 3 |
| 3 | 6.0004201681 | 5.9995798614 | 6.0000000147 | 7 |
| 4 | 6.0000000147 | 5.9999999853 | 6.0000000000 | all 10 shown |
The count of correct decimals went 1, 3, 7 and all 10 over 4 steps — roughly doubling each time — until the guess matched √36 = 6.0000000000 to every decimal shown.
√36 in geometry and everyday measurements
- A square room or garden bed covering 36 square feet measures exactly 6 ft (6 ft) along each wall.
- 36 is not a sum of two whole-number squares — no pair of squares adds up to it — so √36 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 4 box, because 2² + 4² + 4² = 36.
Square roots near √36 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √33 | √33 | 5.7446 | No |
| √34 | √34 | 5.8310 | No |
| √35 | √35 | 5.9161 | No |
| √36 | 6 | 6.0000 | Yes |
| √37 | √37 | 6.0828 | No |
| √38 | √38 | 6.1644 | No |
| √39 | √39 | 6.2450 | No |
- The cube root of 36 is about 3.301927.
- Taking the square root twice gives the fourth root: √6 ≈ 2.449490.
- Four times the radicand doubles the root: √144 = 2 × √36 ≈ 12.
Frequently asked questions
What is the square root of 36?
The square root of 36 is 6, because 6 × 6 = 36. Strictly, 36 has two square roots, 6 and −6; the √ symbol means the positive one.
Is the square root of 36 rational or irrational?
Rational. 36 is a perfect square, so its square root is the whole number 6.
Can √36 be simplified?
Yes, all the way to a whole number: √36 = 6.
What is √36 rounded to two decimal places?
√36 is exactly 6, or 6.00 written to two decimal places.