Solving for the square root of 98 may seem daunting at first, but with some knowledge of basic math principles and a few techniques, it is possible to find the answer.

Firstly, it is important to note that the square root of 98 cannot be simplified into a whole number. Therefore, we must resort to finding an approximate value.

One technique used to approximate the square root of a number is the prime factorization method. To use this method, we break down the number under the radical sign (in this case, 98) into its prime factors, then group them in pairs. We then take one number from each pair and multiply them together to get the closest perfect square. In this case, the prime factorization of 98 is 2 x 7 x 7. Pairing the 7s, we get (2 x 7) x 7 = 14 x 7 = 98. Therefore, we can write the square root of 98 as the square root of (2 x 7 x 7), which equals (7 x the square root of 2).

Another technique used to approximate the square root of a number is the long division method. With this method, we repeatedly divide the number under the radical sign by a divisor that leaves a remainder. This process continues until a desired level of accuracy is achieved. For example, starting with an estimate of 10, we divide 98 by 20, which equals 4 with a remainder of 18. We then bring down the next two digits (00) and add a decimal point, making our quotient 7. We then double our divisor (20) to get 40, and ask how many times 40 goes into 180 (our remainder multiplied by 100). The answer is 4, so we subtract 160 from 180 to get a new remainder of 20. We then repeat the process, bringing down the next two digits (00) and adding another decimal place. Our new quotient is 7.4. We continue this process until we reach a desired level of accuracy.

In conclusion, there are several techniques that can be used to approximate the square root of 98. While it may not be possible to find an exact value, using methods such as prime factorization and long division can give us a close approximation.