I’m not sure what you mean by simplifying a square root.
The definition of the square root of a number we’ll call n is another number, which, when multiplied by itself, equals n. So let’s take an example. The square root of 81 (so n in this case is 81) is 9 because 9 times 9 is 81.
You can also have square roots of a fraction. The square root of 4/9 is 2/3 because the square root of 4 is 2 and the square root of 9 is 3.
But there are many numbers that don’t have such easy square roots. There is a technique for finding the square root that’s a bit like long division, but most people use calculators these days to work out square roots. You can make a guess at a square root. For example, what is the square root of 69? Since the square root of 64 is 8 and the next perfect square is 81, which has a square root of 9, we know that the square root of 69 must be between 8 and 9. The square root of 69 is an irrational number, which means that if we write it out as a decimal number, it starts with 8 but no matter how many decimal places we take the decimal goes on forever. And we can’t use a fraction to express the square root.
But we can approximate to a square root. For example, 10/7 or 1³⁄₇ is an approximation for the square root of 2. Let’s see why 10/7 is a fair approximation. Multiply 10/7 by itself and we get 100/49. Because 10/7 approximates to √2, multiplying it by itself (squarng it) should give us a number close to 2. 100/49 is 98/49+2/49 which is 2 + 2/49 or 2²⁄₄₉. This is a little bit more than 2 (about 2.04), pretty close. And there are various tricks for finding square roots by calculation, but they can be tedious.
