√112 at a glance
- Exact value
- 4√7
- Decimal (10 places)
- 10.5830052443
- Rounded
- 10.6 · 10.58 · 10.583
- Perfect square?
- No — between 10² and 11²
- Rational?
- Irrational
- Both square roots
- ±10.583005
- Prime factorization
- 2⁴ × 7
- Cube root
- 4.820285
How to simplify √112
Look for the largest perfect square that divides 112. Here it is 16 (4²), because 112 = 16 × 7 and 7 has no square factor left:
The prime factorization tells the same story: 112 = 2⁴ × 7. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 7 stays inside.
112 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √112 = 2√28, and √28 can be simplified again. Using 16 straight away finishes in one step.
Check: (4√7)² = 4² × 7 = 16 × 7 = 112. As a decimal, 4√7 = 4 × 2.6457513111 ≈ 10.5830052443.
Where √112 sits between perfect squares
100 = 10² and 121 = 11² are the nearest perfect squares, so √112 lies between 10 and 11. 112 is 12 above 100 and 9 below 121, so the root is closer to 11.
- Straight line between 100 and 121: 10.5714 (0.11% low)
- Tangent from 10, i.e. 10 + 12 ÷ 20: 10.6000 (0.16% high)
- Tangent from 11, i.e. 11 − 9 ÷ 22: 10.5909 (0.07% high)
For √112 the tangent at 11 wins, missing by only 0.0079. Tangent estimates shine when the number sits close to a perfect square — here 112 is just 9 below 121.
Finding √112 with the Babylonian method
Picture a rectangle with an area of 112 and one side x; the other side must be 112 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √112.
Start from the nearest whole number, 11 (11² = 121):
| Step | Guess x | 112 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 11.0000000000 | 10.1818181818 | 10.5909090909 | 2 |
| 2 | 10.5909090909 | 10.5751072961 | 10.5830081935 | 5 |
| 3 | 10.5830081935 | 10.5830022950 | 10.5830052443 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √112 = 10.5830052443 to every decimal shown.
√112 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √112 the pattern is [10; 1, 1, 2, 1, 1, 20] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √112 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 |
|---|---|---|
| 10/1 | 10.0000000000 | 5.8 × 10⁻¹ |
| 11/1 | 11.0000000000 | 4.2 × 10⁻¹ |
| 21/2 | 10.5000000000 | 8.3 × 10⁻² |
| 53/5 | 10.6000000000 | 1.7 × 10⁻² |
| 74/7 | 10.5714285714 | 1.2 × 10⁻² |
| 127/12 | 10.5833333333 | 3.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 112y² = 1. Its smallest solution in positive whole numbers is x = 127, y = 12.
√112 in geometry and everyday measurements
- A square room or garden bed covering 112 square feet measures about 10.58 ft (10 ft 7 in) along each wall.
- 112 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 √112 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 √112 as its space diagonal.
- Since √112 = 4√7, a length of √112 is exactly 4 copies of the length √7 laid end to end.
Square roots near √112 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √109 | √109 | 10.4403 | No |
| √110 | √110 | 10.4881 | No |
| √111 | √111 | 10.5357 | No |
| √112 | 4√7 | 10.5830 | No |
| √113 | √113 | 10.6301 | No |
| √114 | √114 | 10.6771 | No |
| √115 | √115 | 10.7238 | No |
- The cube root of 112 is about 4.820285.
- Four times the radicand doubles the root: √448 = 2 × √112 ≈ 21.16601.
Frequently asked questions
What is the square root of 112?
The square root of 112 is 4√7 in simplest radical form, which is about 10.5830052443. The negative root, −10.583005, also squares to 112.
Is the square root of 112 rational or irrational?
Irrational. 112 is not a perfect square — it falls between 100 and 121 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √112 be simplified?
Yes. The largest perfect square dividing 112 is 16, so √112 = √16 × √7 = 4√7.
What is √112 rounded to two decimal places?
√112 ≈ 10.58 to two decimal places (10.6 to one, 10.583 to three). Check: 10.58² = 111.9364, close to 112.