√900 at a glance
- Exact value
- 30
- Decimal (10 places)
- 30
- Rounded
- 30
- Perfect square?
- Yes — 30²
- Rational?
- Rational (whole number)
- Both square roots
- ±30
- Prime factorization
- 2² × 3² × 5²
- Cube root
- 9.654894
How to simplify √900
Write 900 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√900 = 2 × 3 × 5 = 30
900 has 14 factor pairs (1 × 900, 2 × 450, 3 × 300, 4 × 225, 5 × 180, 6 × 150, 9 × 100, 10 × 90, 12 × 75, 15 × 60, 18 × 50, 20 × 45, 25 × 36, 30 × 30). The pair with two equal numbers, 30 × 30, 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 900 = 1 + 3 + 5 + 7 + … + 59, the first 30 odd numbers.
Where √900 sits between perfect squares
900 is 30². The perfect squares on either side are 841 (29²) and 961 (31²), so the gaps are 59 below and 61 above — consecutive squares always differ by consecutive odd numbers.
Finding √900 with the Babylonian method
Picture a rectangle with an area of 900 and one side x; the other side must be 900 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √900.
To show the method at work on a perfect square, start one above the answer, at 31:
| Step | Guess x | 900 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 29.0322580645 | 30.0161290323 | 1 |
| 2 | 30.0161290323 | 29.9838796346 | 30.0000043334 | 5 |
| 3 | 30.0000043334 | 29.9999956666 | 30.0000000000 | all 10 shown |
The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √900 = 30.0000000000 to every decimal shown.
√900 in geometry and everyday measurements
- 900 square feet is 83.6 m². Laid out as a square — a small house footprint or a lot — it is exactly 30 ft (30 ft) on a side.
- 900 = 18² + 24², so 18, 24 and 30 form a Pythagorean triple: a right triangle with legs 18 and 24 has a hypotenuse of exactly 30.
Square roots near √900 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √897 | √897 | 29.9500 | No |
| √898 | √898 | 29.9666 | No |
| √899 | √899 | 29.9833 | No |
| √900 | 30 | 30.0000 | Yes |
| √901 | √901 | 30.0167 | No |
| √902 | √902 | 30.0333 | No |
| √903 | √903 | 30.0500 | No |
- The cube root of 900 is about 9.654894.
- Taking the square root twice gives the fourth root: √30 ≈ 5.477226.
- Dividing by 100 divides the root by 10: √9 = √900 ÷ 10 ≈ 3.
Frequently asked questions
What is the square root of 900?
The square root of 900 is 30, because 30 × 30 = 900. Strictly, 900 has two square roots, 30 and −30; the √ symbol means the positive one.
Is the square root of 900 rational or irrational?
Rational. 900 is a perfect square, so its square root is the whole number 30.
Can √900 be simplified?
Yes, all the way to a whole number: √900 = 30.
What is √900 rounded to two decimal places?
√900 is exactly 30, or 30.00 written to two decimal places.