√847 at a glance
- Exact value
- 11√7
- Decimal (10 places)
- 29.1032644217
- Rounded
- 29.1 · 29.10 · 29.103
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.103264
- Prime factorization
- 7 × 11²
- Cube root
- 9.461525
How to simplify √847
Look for the largest perfect square that divides 847. Here it is 121 (11²), because 847 = 121 × 7 and 7 has no square factor left:
The prime factorization tells the same story: 847 = 7 × 11². Each pair of equal primes leaves the radical as one factor, so 11 comes out and 7 stays inside.
Check: (11√7)² = 11² × 7 = 121 × 7 = 847. As a decimal, 11√7 = 11 × 2.6457513111 ≈ 29.1032644217.
Where √847 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √847 lies between 29 and 30. 847 is 6 above 841 and 53 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.1017 (0.01% low)
- Tangent from 29, i.e. 29 + 6 ÷ 58: 29.1034 (0% high)
- Tangent from 30, i.e. 30 − 53 ÷ 60: 29.1167 (0.05% high)
For √847 the tangent at 29 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 847 is just 6 above 841.
Finding √847 with the Babylonian method
If a guess is too big, 847 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√847) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 847 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.2068965517 | 29.1034482759 | 3 |
| 2 | 29.1034482759 | 29.1030805687 | 29.1032644223 | 9 |
| 3 | 29.1032644223 | 29.1032644211 | 29.1032644217 | all 10 shown |
The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √847 = 29.1032644217 to every decimal shown.
√847 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √847 the pattern is [29; 9, 1, 2, 6, 8, 6, 2, 1, 9, 58] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √847 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 29/1 | 29.0000000000 | 1.0 × 10⁻¹ |
| 262/9 | 29.1111111111 | 7.8 × 10⁻³ |
| 291/10 | 29.1000000000 | 3.3 × 10⁻³ |
| 844/29 | 29.1034482759 | 1.8 × 10⁻⁴ |
| 5,355/184 | 29.1032608696 | 3.6 × 10⁻⁶ |
| 43,684/1,501 | 29.1032644903 | 6.9 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 847y² = 1. Its smallest solution in positive whole numbers is x = 8,193,151, y = 281,520.
√847 in geometry and everyday measurements
- 847 square feet is 78.7 m². Laid out as a square — a small house footprint or a lot — it is about 29.1 ft (29 ft 1 in) on a side.
- 847 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √847 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √847 as its space diagonal.
- Since √847 = 11√7, a length of √847 is exactly 11 copies of the length √7 laid end to end.
Square roots near √847 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √844 | 2√211 | 29.0517 | No |
| √845 | 13√5 | 29.0689 | No |
| √846 | 3√94 | 29.0861 | No |
| √847 | 11√7 | 29.1033 | No |
| √848 | 4√53 | 29.1204 | No |
| √849 | √849 | 29.1376 | No |
| √850 | 5√34 | 29.1548 | No |
- The cube root of 847 is about 9.461525.
- Squaring undoes the root: (√847)² = 847, while 847² = 717,409 — the number whose square root is 847.
Frequently asked questions
What is the square root of 847?
The square root of 847 is 11√7 in simplest radical form, which is about 29.1032644217. The negative root, −29.103264, also squares to 847.
Is the square root of 847 rational or irrational?
Irrational. 847 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √847 be simplified?
Yes. The largest perfect square dividing 847 is 121, so √847 = √121 × √7 = 11√7.
What is √847 rounded to two decimal places?
√847 ≈ 29.10 to two decimal places (29.1 to one, 29.103 to three). Check: 29.10² = 846.81, close to 847.