√848 at a glance
- Exact value
- 4√53
- Decimal (10 places)
- 29.1204395571
- Rounded
- 29.1 · 29.12 · 29.120
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.120440
- Prime factorization
- 2⁴ × 53
- Cube root
- 9.465247
How to simplify √848
Look for the largest perfect square that divides 848. Here it is 16 (4²), because 848 = 16 × 53 and 53 has no square factor left:
The prime factorization tells the same story: 848 = 2⁴ × 53. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 53 stays inside.
848 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √848 = 2√212, and √212 can be simplified again. Using 16 straight away finishes in one step.
Check: (4√53)² = 4² × 53 = 16 × 53 = 848. As a decimal, 4√53 = 4 × 7.2801098893 ≈ 29.1204395571.
Where √848 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √848 lies between 29 and 30. 848 is 7 above 841 and 52 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.1186 (0.01% low)
- Tangent from 29, i.e. 29 + 7 ÷ 58: 29.1207 (0% high)
- Tangent from 30, i.e. 30 − 52 ÷ 60: 29.1333 (0.04% high)
For √848 the tangent at 29 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 848 is just 7 above 841.
Finding √848 with the Babylonian method
Picture a rectangle with an area of 848 and one side x; the other side must be 848 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √848.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 848 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.2413793103 | 29.1206896552 | 3 |
| 2 | 29.1206896552 | 29.1201894612 | 29.1204395582 | 8 |
| 3 | 29.1204395582 | 29.1204395560 | 29.1204395571 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √848 = 29.1204395571 to every decimal shown.
√848 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √848 the pattern is [29; 8, 3, 3, 3, 8, 58] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √848 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 | 1.2 × 10⁻¹ |
| 233/8 | 29.1250000000 | 4.6 × 10⁻³ |
| 728/25 | 29.1200000000 | 4.4 × 10⁻⁴ |
| 2,417/83 | 29.1204819277 | 4.2 × 10⁻⁵ |
| 7,979/274 | 29.1204379562 | 1.6 × 10⁻⁶ |
| 66,249/2,275 | 29.1204395604 | 3.3 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 848y² = 1. Its smallest solution in positive whole numbers is x = 66,249, y = 2,275.
√848 in geometry and everyday measurements
- 848 square feet is 78.8 m². Laid out as a square — a small house footprint or a lot — it is about 29.12 ft (29 ft 1 in) on a side.
- 848 = 8² + 28², so by the Pythagorean theorem √848 is the diagonal of a 8 × 28 rectangle — and the distance between the points (0, 0) and (8, 28) on a grid.
- Since √848 = 4√53, a length of √848 is exactly 4 copies of the length √53 laid end to end.
Square roots near √848 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √851 | √851 | 29.1719 | No |
- The cube root of 848 is about 9.465247.
- Because 848 = 4 × 212, the root is twice √212: 2 × 14.56022 ≈ 29.12044.
Frequently asked questions
What is the square root of 848?
The square root of 848 is 4√53 in simplest radical form, which is about 29.1204395571. The negative root, −29.120440, also squares to 848.
Is the square root of 848 rational or irrational?
Irrational. 848 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 √848 be simplified?
Yes. The largest perfect square dividing 848 is 16, so √848 = √16 × √53 = 4√53.
What is √848 rounded to two decimal places?
√848 ≈ 29.12 to two decimal places (29.1 to one, 29.120 to three). Check: 29.12² = 847.9744, close to 848.