√845 at a glance
- Exact value
- 13√5
- Decimal (10 places)
- 29.0688837075
- Rounded
- 29.1 · 29.07 · 29.069
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.068884
- Prime factorization
- 5 × 13²
- Cube root
- 9.454072
How to simplify √845
Look for the largest perfect square that divides 845. Here it is 169 (13²), because 845 = 169 × 5 and 5 has no square factor left:
The prime factorization tells the same story: 845 = 5 × 13². Each pair of equal primes leaves the radical as one factor, so 13 comes out and 5 stays inside.
Check: (13√5)² = 13² × 5 = 169 × 5 = 845. As a decimal, 13√5 = 13 × 2.2360679775 ≈ 29.0688837075.
Where √845 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √845 lies between 29 and 30. 845 is 4 above 841 and 55 below 900, so the root is closer to 29.
- Straight line between 841 and 900: 29.0678 (0% low)
- Tangent from 29, i.e. 29 + 4 ÷ 58: 29.0690 (0% high)
- Tangent from 30, i.e. 30 − 55 ÷ 60: 29.0833 (0.05% high)
For √845 the tangent at 29 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 845 is just 4 above 841.
Finding √845 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 845: following the tangent line down to zero simplifies to averaging x with 845 ÷ x.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 845 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 29.1379310345 | 29.0689655172 | 4 |
| 2 | 29.0689655172 | 29.0688018980 | 29.0688837076 | 9 |
| 3 | 29.0688837076 | 29.0688837074 | 29.0688837075 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √845 = 29.0688837075 to every decimal shown.
√845 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √845 the pattern is [29; 14, 1, 1, 14, 58] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √845 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 | 6.9 × 10⁻² |
| 407/14 | 29.0714285714 | 2.5 × 10⁻³ |
| 436/15 | 29.0666666667 | 2.2 × 10⁻³ |
| 843/29 | 29.0689655172 | 8.2 × 10⁻⁵ |
| 12,238/421 | 29.0688836105 | 9.7 × 10⁻⁸ |
| 710,647/24,447 | 29.0688837076 | 1.2 × 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 845y² = 1. Its smallest solution in positive whole numbers is x = 299,537,289, y = 10,304,396. Because the period is odd, the equation with −1 on the right also has a solution: 12,238² − 845 × 421² = −1.
√845 in geometry and everyday measurements
- 845 square feet is 78.5 m². Laid out as a square — a small house footprint or a lot — it is about 29.07 ft (29 ft 1 in) on a side.
- 845 = 2² + 29² = 13² + 26² = 19² + 22², so by the Pythagorean theorem √845 is the diagonal of rectangles measuring 2 × 29, 13 × 26 and 19 × 22 — and the distance between the points (0, 0) and (2, 29) on a grid.
- Since √845 = 13√5, a length of √845 is exactly 13 copies of the length √5 laid end to end.
Square roots near √845 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √842 | √842 | 29.0172 | No |
| √843 | √843 | 29.0345 | No |
| √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 |
- The cube root of 845 is about 9.454072.
- Squaring undoes the root: (√845)² = 845, while 845² = 714,025 — the number whose square root is 845.
Frequently asked questions
What is the square root of 845?
The square root of 845 is 13√5 in simplest radical form, which is about 29.0688837075. The negative root, −29.068884, also squares to 845.
Is the square root of 845 rational or irrational?
Irrational. 845 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 √845 be simplified?
Yes. The largest perfect square dividing 845 is 169, so √845 = √169 × √5 = 13√5.
What is √845 rounded to two decimal places?
√845 ≈ 29.07 to two decimal places (29.1 to one, 29.069 to three). Check: 29.07² = 845.0649, close to 845.