156223

This problem starts with adding surds and when we add surds we want a whole number denominator and with surds the only way to do that is to rationalise the denomintator by inversing the sign in the middle of the surds: if it was sqaure root of 1 + square root of 2 the rationalise version of it would be sqaure root of 1 - the square root of 2 on the denominator and you have to do the same to the top and bottom so you have to times the top by what your rationalising the bottom by, Then after this you will see that the bottom will always be -1 and the top the thing thing your rationlising by so when you write it it will be -(sqaure of 1 - sqaure of 2) when you expand it the signs will flip and when you do that to the next one the sqaure roots will always cancel each other out apart from your first answer and your last answer which are sqaure root of -(1) and sqaure root of 100 which is -1 +10=9 and the answer is a whole number which is what she found surprising.