I read in manhattan GMAT book as even root of a number results only +ve but not -ve in GMAT.
example:-
if x=sqrt64 then x=8 but not -8

Many other books and material always mentions like
if x^2=64 then x=+1/-1

which is correct? did I misunderstand any?

Those are both correct.

THE SQUARE ROOT SYMBOL ITSELF carries the implicit meaning of choosing the positive root. This is so because symbols - which represent FUNCTIONS - can't take on more than one value; therefore, the symbol is DEFINED to take on the positive ('more useful') value.

In the equation x^2 = 64, there is no use of the square root symbol. This is an EQUATION which is allowed to have multiple solutions (it's not a FUNCTION of x). Therefore, we are after ALL x's that will solve it (two of them, in this case).

By the way, I certainly hope that your books aren't telling you that 1 and -1 are solutions to x^2 = 64 (the correct answers, of course, are 8 and -8). If they are, oh dear...



PS If you know trigonometry, this is the same reason why arcsin(1/2) is ONLY defined to be 30 degrees, but tons of x's solve the equation sin x = 1/2 (like 150, 390, -330, and so on). If you don't know trigonometry, then ignore this example, but pay attention to the text above.