√441 at a glance
- Exact value
- 21
- Decimal (10 places)
- 21
- Rounded
- 21
- Perfect square?
- Yes — 21²
- Rational?
- Rational (whole number)
- Both square roots
- ±21
- Prime factorization
- 3² × 7²
- Cube root
- 7.611663
How to simplify √441
Write 441 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√441 = 3 × 7 = 21
441 has 5 factor pairs (1 × 441, 3 × 147, 7 × 63, 9 × 49, 21 × 21). The pair with two equal numbers, 21 × 21, 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 441 = 1 + 3 + 5 + 7 + … + 41, the first 21 odd numbers.
Where √441 sits between perfect squares
441 is 21². The perfect squares on either side are 400 (20²) and 484 (22²), so the gaps are 41 below and 43 above — consecutive squares always differ by consecutive odd numbers.
Finding √441 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 441: following the tangent line down to zero simplifies to averaging x with 441 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 22:
| Step | Guess x | 441 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 22.0000000000 | 20.0454545455 | 21.0227272727 | 1 |
| 2 | 21.0227272727 | 20.9772972973 | 21.0000122850 | 4 |
| 3 | 21.0000122850 | 20.9999877150 | 21.0000000000 | 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 √441 = 21.0000000000 to every decimal shown.
√441 in geometry and everyday measurements
- A square garage floor of 441 square feet measures exactly 21 ft (21 ft) per side, and its corner-to-corner diagonal is √882 ≈ 29.7 ft.
- 441 is not a sum of two whole-number squares — no pair of squares adds up to it — so √441 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 5 × 20 box, because 4² + 5² + 20² = 441.
Square roots near √441 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √438 | √438 | 20.9284 | No |
| √439 | √439 | 20.9523 | No |
| √440 | 2√110 | 20.9762 | No |
| √441 | 21 | 21.0000 | Yes |
| √442 | √442 | 21.0238 | No |
| √443 | √443 | 21.0476 | No |
| √444 | 2√111 | 21.0713 | No |
- The cube root of 441 is about 7.611663.
- Taking the square root twice gives the fourth root: √21 ≈ 4.582576.
- Squaring undoes the root: (√441)² = 441, while 441² = 194,481 — the number whose square root is 441.
Frequently asked questions
What is the square root of 441?
The square root of 441 is 21, because 21 × 21 = 441. Strictly, 441 has two square roots, 21 and −21; the √ symbol means the positive one.
Is the square root of 441 rational or irrational?
Rational. 441 is a perfect square, so its square root is the whole number 21.
Can √441 be simplified?
Yes, all the way to a whole number: √441 = 21.
What is √441 rounded to two decimal places?
√441 is exactly 21, or 21.00 written to two decimal places.