√196 at a glance
- Exact value
- 14
- Decimal (10 places)
- 14
- Rounded
- 14
- Perfect square?
- Yes — 14²
- Rational?
- Rational (whole number)
- Both square roots
- ±14
- Prime factorization
- 2² × 7²
- Cube root
- 5.808786
How to simplify √196
Write 196 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√196 = 2 × 7 = 14
196 has 5 factor pairs (1 × 196, 2 × 98, 4 × 49, 7 × 28, 14 × 14). The pair with two equal numbers, 14 × 14, 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 196 = 1 + 3 + 5 + 7 + … + 27, the first 14 odd numbers.
Where √196 sits between perfect squares
196 is 14². The perfect squares on either side are 169 (13²) and 225 (15²), so the gaps are 27 below and 29 above — consecutive squares always differ by consecutive odd numbers.
Finding √196 with the Babylonian method
Picture a rectangle with an area of 196 and one side x; the other side must be 196 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √196.
To show the method at work on a perfect square, start one above the answer, at 15:
| Step | Guess x | 196 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 15.0000000000 | 13.0666666667 | 14.0333333333 | 1 |
| 2 | 14.0333333333 | 13.9667458432 | 14.0000395883 | 4 |
| 3 | 14.0000395883 | 13.9999604118 | 14.0000000001 | 10 |
| 4 | 14.0000000001 | 13.9999999999 | 14.0000000000 | all 10 shown |
The count of correct decimals went 1, 4, 10 and all 10 over 4 steps — roughly doubling each time — until the guess matched √196 = 14.0000000000 to every decimal shown.
√196 in geometry and everyday measurements
- A square room or garden bed covering 196 square feet measures exactly 14 ft (14 ft) along each wall.
- 196 is not a sum of two whole-number squares — no pair of squares adds up to it — so √196 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 6 × 12 box, because 4² + 6² + 12² = 196.
Square roots near √196 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √193 | √193 | 13.8924 | No |
| √194 | √194 | 13.9284 | No |
| √195 | √195 | 13.9642 | No |
| √196 | 14 | 14.0000 | Yes |
| √197 | √197 | 14.0357 | No |
| √198 | 3√22 | 14.0712 | No |
| √199 | √199 | 14.1067 | No |
- The cube root of 196 is about 5.808786.
- Taking the square root twice gives the fourth root: √14 ≈ 3.741657.
- Four times the radicand doubles the root: √784 = 2 × √196 ≈ 28.
Frequently asked questions
What is the square root of 196?
The square root of 196 is 14, because 14 × 14 = 196. Strictly, 196 has two square roots, 14 and −14; the √ symbol means the positive one.
Is the square root of 196 rational or irrational?
Rational. 196 is a perfect square, so its square root is the whole number 14.
Can √196 be simplified?
Yes, all the way to a whole number: √196 = 14.
What is √196 rounded to two decimal places?
√196 is exactly 14, or 14.00 written to two decimal places.