`((x^(p))/(x^(q)))^((p+q))((x^(q))/(x^(r)))^((q+r))((x^(r))/(x^(p)))^((r+p))`

Simply (a^(p)/a^(q))^(p+q-r).(a^(q)/a^(r ))^(q+r-p).(a^(r )/a^(p))^(r+p-q)

x^(p-q).x^(q-r).x^(r-p)

Prove root(p+q)(x^(p^(2))/(x^(q^(2))))xxroot(q+r)(x^(q^(2))/(x^(r^(2))))xxroot(r+p)(x^(r^(2))/(x^(p^(2))))=1

sqrt(x^(p-q))sqrt(x^(q-r))sqrt(x^(r-p))=1

If y = (1)/(1+x^( p-r) +x^(q-r)) +(1)/( 1+ x^(p-q) +x^(r-q) )+ (1) /( 1+x^(q-p) +x^(r-p)),then (dy)/(dx) =

If P -= ((1)/(x_(p)), p ), Q = ((1)/(x_(q)),q), R = ((1)/(x_(r)), r) where x _(k) ne 0 , denotes the k^(th) terms of a H.P. for k in N , then :