Find x^2y^2+y^2z^2+x^2z^2/X^4+y^4+Z^4 if x+y+z=0

Question

Accepted Answer

i f space x plus y plus z equals 0 space t h e n space x squared plus y squared plus z squared equals minus 2 open parentheses x y plus y z plus z x close parentheses s q u a r i n g space i t space a g a i n space g i v e s open parentheses space x squared plus y squared plus z squared close parentheses squared equals 4 open parentheses x y plus y z plus z x close parentheses squared x to the power of 4 plus y to the power of 4 plus z to the power of 4 plus 2 left parenthesis x squared y squared plus y squared z squared plus z squared x squared right parenthesis equals 4 open parentheses x squared y squared plus y squared z squared plus z squared x squared plus 2 x y z open parentheses x plus y plus z close parentheses close parentheses x to the power of 4 plus y to the power of 4 plus z to the power of 4 equals 2 left parenthesis x squared y squared plus y squared z squared plus z squared x squared right parenthesis space space open square brackets because x plus y plus z equals 0 close square brackets fraction numerator left parenthesis x squared y squared plus y squared z squared plus z squared x squared right parenthesis over denominator x to the power of 4 plus y to the power of 4 plus z to the power of 4 end fraction equals 1 half