(1)/(a+b+x)=(1)/(a)+(1)/(b)+(1)/(x)

(1)/(x+a)+(1)/(x+b)=(1)/(x+a+b)+(1)/(x)

(x^((1)/(a-b)))^((1)/(a-c))(x^((1)/(b-c)))^((1)/(b-a))((1)/(x^(c-a)))^((1)/(c-b))

(x^((1)/(a-b)))^((1)/(a-c))xx(x^((1)/(b-c)))^((1)/(b-a))xx(x^((1)/(c-a)))^((1)/(c-b))

Prove that. (i) sqrt(x^(-1) y) .sqrt(y^(-1) z) . Sqrt(z^(-1) x) = 1 (ii) ((1)/(x^(a-b)))^((1)/(a-c)).((1)/(x^(b-c))).((1)/(x^(c-b)))^((1)/(c-b))= 1 (iii) (x^(a(b-c)))/(x^(b(a-c))) div ((x^(b))/(x^(a))) (iv) ((x^(a+b))^(2)(x^(b+c))^(2)(x^(c+a))^(2))/((x^(a)x^(b)x^(c))^(4))

Expression ((a+1/b)^(1/x)(a-1/b)^(1/x))/((b+1/a)^(1/x)(b-1/a)^(1/x)) is an integer if

Simply (x^(1/(a-b)))^(1/(a-c))xx(x^(1/(b-a)))^(1/(b-c))xx(x^(1/(c -a)))^(1/(c-b))

(x^(1/(a-b)))^(1/(a-c))xx(x^(1/(b-c)))^(1/(b-a))xx(x^(1/(c-a)))^(1/(c-b))

(x^(1/(a-b)))^(1/(a-c))xx(x^(1/(b-c)))^(1/(b-a))xx(x^(1/(c-a)))^(1/(c-b))

Expression ((a+1/b)^(1/x)*(a-1/b)^(1/x))/((b+1/a)^(1/x)*(b-1/a)^(1/x)) is an integer if

(1)/(x^(b)+x^(-c)+1)+(1)/(x^(c)+x^(-a)+1)+(1)/(x^(a)+x^(-b)+1)