Finding square roots come in almost every mathematics syllabus from 8th class. Its simple to find the square root of perfect square that is one which does not give a decimal square root. But there are others that are not perfect squares. These are surds.

Square roots of perfect squares can be found easily by factorization. But such a method cannot be used for finding square root of surds. We can find the root of 4 or 144 with the help of factorization but we need the help of long division method to find square root of surds (for example root of 2, 3, 5 etc.).

The method of doing square root can be learnt by visiting https://www.math-only-math.com/square-root-of-numbers-that-are-not-perfect-squares.html

Hello.

Let √(19+6√2) = √a + √ b

Squaring both sides

{√(19+6√2)}^2 = (√a + √ b)^2

19 +6√2 = a+b +2√ab

19 +2√(3x3x2) = a+b +2√ab

Equating rational and irrational parts we get a+b = 19 and ab = 18

By intuition a= 18 and b = 1

So, √(19+6√2) = √18 + √ 1 = 3√2 + 1