√400 at a glance
- Exact value
- 20
- Decimal (10 places)
- 20
- Rounded
- 20
- Perfect square?
- Yes — 20²
- Rational?
- Rational (whole number)
- Both square roots
- ±20
- Prime factorization
- 2⁴ × 5²
- Cube root
- 7.368063
How to simplify √400
Write 400 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√400 = 2² × 5 = 20
400 has 8 factor pairs (1 × 400, 2 × 200, 4 × 100, 5 × 80, 8 × 50, 10 × 40, 16 × 25, 20 × 20). The pair with two equal numbers, 20 × 20, 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 400 = 1 + 3 + 5 + 7 + … + 39, the first 20 odd numbers.
Where √400 sits between perfect squares
400 is 20². The perfect squares on either side are 361 (19²) and 441 (21²), so the gaps are 39 below and 41 above — consecutive squares always differ by consecutive odd numbers.
Finding √400 with the Babylonian method
Picture a rectangle with an area of 400 and one side x; the other side must be 400 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √400.
To show the method at work on a perfect square, start one above the answer, at 21:
| Step | Guess x | 400 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 19.0476190476 | 20.0238095238 | 1 |
| 2 | 20.0238095238 | 19.9762187872 | 20.0000141555 | 4 |
| 3 | 20.0000141555 | 19.9999858445 | 20.0000000000 | all 10 shown |
The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √400 = 20.0000000000 to every decimal shown.
√400 in geometry and everyday measurements
- A square patio or deck of 400 square feet is exactly 20 ft (20 ft) on each side, so edging all the way around takes 4 × √400 ≈ 80 ft.
- 400 = 12² + 16², so 12, 16 and 20 form a Pythagorean triple: a right triangle with legs 12 and 16 has a hypotenuse of exactly 20.
Square roots near √400 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √397 | √397 | 19.9249 | No |
| √398 | √398 | 19.9499 | No |
| √399 | √399 | 19.9750 | No |
| √400 | 20 | 20.0000 | Yes |
| √401 | √401 | 20.0250 | No |
| √402 | √402 | 20.0499 | No |
| √403 | √403 | 20.0749 | No |
- The cube root of 400 is about 7.368063.
- Taking the square root twice gives the fourth root: √20 ≈ 4.472136.
- Dividing by 100 divides the root by 10: √4 = √400 ÷ 10 ≈ 2.
Frequently asked questions
What is the square root of 400?
The square root of 400 is 20, because 20 × 20 = 400. Strictly, 400 has two square roots, 20 and −20; the √ symbol means the positive one.
Is the square root of 400 rational or irrational?
Rational. 400 is a perfect square, so its square root is the whole number 20.
Can √400 be simplified?
Yes, all the way to a whole number: √400 = 20.
What is √400 rounded to two decimal places?
√400 is exactly 20, or 20.00 written to two decimal places.