Here,
#root(4)(194-x)+root(4)x=4#
Let, #color(red)a=root(4)(194-x)# #andcolor(red)
b=root(4)x=>color(red)(a^4)=194-x andcolor(red)( b^4)=x#
#:.color(blue)(a+b=4...to(I) and a^4+b^4=194...to(II)#
Squaring #(I)=>(a+b)^2=4^2=>a^2+2ab+b^2=16#
i.e. #color(blue)(a^2+b^2=16-2ab...to(III)#
From #(II)toa^4+b^4=194#,
#=>color(brown)((a^2+b^2)^2-2a^2b^2=194#
#=>(16-2ab)^2-2a^2b^2=194...to#using #(III)#
#=>256-64ab+4a^2b^2-2a^2b^2=194#
#=>2a^2b^2-64ab+256=194#
#=>a^2b^2-32ab+128=97#
#=>a^2b^2-32ab+256=97+128=225#
#=>(ab-16)^2=15^2#
#=>ab-16=+-15#
#=>ab=16+-15=>color(blue)(ab=1 or ab=31#
If #ab=1,then ,a=1/b to# #""color(red)"Please see the comment below"#
From #(I)to1/b+b=4=>1+b^2=4b#
#=>b^2-4b+1=0=>b^2-4b+4=3=>(b-2)^2=(sqrt3)^2#
#=>b-2=+-sqrt3=>b=2+-sqrt3#
#b^2=(2+-sqrt3)^2=4+-4sqrt3+3=7+-4sqrt3#
#b^4=(7+-4sqrt3)^2=49+-56sqrt3+48#
#color(red)(x=97+-56sqrt3#
Note:If #ab=31=>a=31/b#
From #(I)to31/b+b=4=>b^2-4b+31=0#
#triangle=16-124<0=>color(blue)(ab!=31#
