√100 at a glance
- Exact value
- 10
- Decimal (10 places)
- 10
- Rounded
- 10
- Perfect square?
- Yes — 10²
- Rational?
- Rational (whole number)
- Both square roots
- ±10
- Prime factorization
- 2² × 5²
- Cube root
- 4.641589
How to simplify √100
Write 100 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√100 = 2 × 5 = 10
100 has 5 factor pairs (1 × 100, 2 × 50, 4 × 25, 5 × 20, 10 × 10). The pair with two equal numbers, 10 × 10, 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 100 = 1 + 3 + 5 + 7 + … + 19, the first 10 odd numbers.
Where √100 sits between perfect squares
100 is 10². The perfect squares on either side are 81 (9²) and 121 (11²), so the gaps are 19 below and 21 above — consecutive squares always differ by consecutive odd numbers.
Finding √100 with the Babylonian method
Picture a rectangle with an area of 100 and one side x; the other side must be 100 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √100.
To show the method at work on a perfect square, start one above the answer, at 11:
| Step | Guess x | 100 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 9.0909090909 | 10.0454545455 | 1 |
| 2 | 10.0454545455 | 9.9547511312 | 10.0001028383 | 3 |
| 3 | 10.0001028383 | 9.9998971627 | 10.0000000005 | 9 |
| 4 | 10.0000000005 | 9.9999999995 | 10.0000000000 | all 10 shown |
The count of correct decimals went 1, 3, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √100 = 10.0000000000 to every decimal shown.
√100 in geometry and everyday measurements
- A square room or garden bed covering 100 square feet measures exactly 10 ft (10 ft) along each wall.
- 100 = 6² + 8², so 6, 8 and 10 form a Pythagorean triple: a right triangle with legs 6 and 8 has a hypotenuse of exactly 10.
Because 100 = 10², every square root of 100 × k is ten times the root of k: √300 = 10√3, √500 = 10√5, and so on.
Square roots near √100 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √97 | √97 | 9.8489 | No |
| √98 | 7√2 | 9.8995 | No |
| √99 | 3√11 | 9.9499 | No |
| √100 | 10 | 10.0000 | Yes |
| √101 | √101 | 10.0499 | No |
| √102 | √102 | 10.0995 | No |
| √103 | √103 | 10.1489 | No |
- The cube root of 100 is about 4.641589.
- Taking the square root twice gives the fourth root: √10 ≈ 3.162278.
- Dividing by 100 divides the root by 10: √1 = √100 ÷ 10 ≈ 1.
Frequently asked questions
What is the square root of 100?
The square root of 100 is 10, because 10 × 10 = 100. Strictly, 100 has two square roots, 10 and −10; the √ symbol means the positive one.
Is the square root of 100 rational or irrational?
Rational. 100 is a perfect square, so its square root is the whole number 10.
Can √100 be simplified?
Yes, all the way to a whole number: √100 = 10.
What is √100 rounded to two decimal places?
√100 is exactly 10, or 10.00 written to two decimal places.