x的y次等于y的x次,y等于9x求y

题目

x的y次等于y的x次,y等于9x求y

答案

x^y=y^x
x^(9x)=(9x)^x
(x^9)^x=(9x)^x
x^9=9x
x(x^8 -9)=0
x(x^4+3)(x²+√3)[x+3^(1/4)][x-3^(1/4)]=0
x^4+3、x²+√3均恒>0,因此只有
x=0 x=-3^(1/4) x=3^(1/4)
x=0时,y=9x=0
x=-3^(1/4)时,y=9x=9×[-3^(1/4)]=-3^(2+1/4)=-3^(9/4)
x=3^(1/4)时,y=9x=9×3^(1/4)=3^(2+1/4)=3^(9/4)