`sqrt(2^x(4^x(0. 125)^(1/ x))^(1/3))=4.(2^(1/3))`

Let 0

lim_(x->a){[(a^(1/2)+x^(1/2))/(a^(1/4)-x^(1/4)))^(- 1)-(2(a x)^(1/4))/(x^(3/4)-a^(1/4)x^(1/2)+a^(1/2)x^(1/4)-a^(3/4))]^(- 1)-sqrt2^(log_4 a)}^8

int_(-(1)/(sqrt(3)))^(-(1)/(sqrt(3)))(x^(4))/(1-x^(4))cos^(-1)((2x)/(1+x^(2)))dx

If x=sqrt(3)+sqrt(2), then x+(1)/(x),x^(2)+(1)/(x^(2)),x^(3)+(1)/(x^(3)),x^(4)+(1)/(x^(4))

If int_((-1)/(sqrt3))^(1//sqrt3)(x^(4))/(1-x^(4))cos^(-1)((2x)/(1+x^(2)))dx=k , then int_(0)^(1//sqrt3)(x^(4))/(1-x^(4))dx the value of k is equal to

lim_(x rarr a){[(a^((1)/(2))+x^((1)/(2)))/(a^((1)/(4))-x^((1)/(4))))^(-1)-(2(ax)^((1)/(4)))/(x^((3)/(4))-a^((1)/(4))x^((1)/(2))+a^((1)/(2))x^((1)/(4))-a^((3)/(4)))]^(-1)-sqrt(2)^(log_(4)a)}^(8)

The value of f(x,y)=((4sqrt(x^(3)y)-4sqrt(x^(3)))/(sqrt(y)-sqrt(x))+(1+sqrt(xy))/(4sqrt(xy)))^(-2)(1+2sqrt((y)/(x))+(y)/(x))^((1)/(2)) when x=9,y=0.04