√841 at a glance
- Exact value
- 29
- Decimal (10 places)
- 29
- Rounded
- 29
- Perfect square?
- Yes — 29²
- Rational?
- Rational (whole number)
- Both square roots
- ±29
- Prime factorization
- 29²
- Cube root
- 9.439131
How to simplify √841
Write 841 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√841 = 29 = 29
841 has 2 factor pairs (1 × 841, 29 × 29). The pair with two equal numbers, 29 × 29, 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 841 = 1 + 3 + 5 + 7 + … + 57, the first 29 odd numbers.
Where √841 sits between perfect squares
841 is 29². The perfect squares on either side are 784 (28²) and 900 (30²), so the gaps are 57 below and 59 above — consecutive squares always differ by consecutive odd numbers.
Finding √841 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 841: following the tangent line down to zero simplifies to averaging x with 841 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 30:
| Step | Guess x | 841 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 28.0333333333 | 29.0166666667 | 1 |
| 2 | 29.0166666667 | 28.9833429064 | 29.0000047865 | 5 |
| 3 | 29.0000047865 | 28.9999952135 | 29.0000000000 | 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 √841 = 29.0000000000 to every decimal shown.
√841 in geometry and everyday measurements
- 841 square feet is 78.1 m². Laid out as a square — a small house footprint or a lot — it is exactly 29 ft (29 ft) on a side.
- 841 = 20² + 21², so 20, 21 and 29 form a Pythagorean triple: a right triangle with legs 20 and 21 has a hypotenuse of exactly 29.
Square roots near √841 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √838 | √838 | 28.9482 | No |
| √839 | √839 | 28.9655 | No |
| √840 | 2√210 | 28.9828 | No |
| √841 | 29 | 29.0000 | Yes |
| √842 | √842 | 29.0172 | No |
| √843 | √843 | 29.0345 | No |
| √844 | 2√211 | 29.0517 | No |
- The cube root of 841 is about 9.439131.
- Taking the square root twice gives the fourth root: √29 ≈ 5.385165.
- Squaring undoes the root: (√841)² = 841, while 841² = 707,281 — the number whose square root is 841.
Frequently asked questions
What is the square root of 841?
The square root of 841 is 29, because 29 × 29 = 841. Strictly, 841 has two square roots, 29 and −29; the √ symbol means the positive one.
Is the square root of 841 rational or irrational?
Rational. 841 is a perfect square, so its square root is the whole number 29.
Can √841 be simplified?
Yes, all the way to a whole number: √841 = 29.
What is √841 rounded to two decimal places?
√841 is exactly 29, or 29.00 written to two decimal places.