Simplify `(p^(a)/p^(b))^(c)xx(p^(b)/p^(c))xx(p^(c)/p^(a))`

Simplify ((p^(a+b))^(5).(p^(b+c))^(5).(p^(c+a))^(5))/((p^(a). p^(b). p^(c))^(10))

The expression ((log_((a)/(b))p)^(2)+(log_((b)/(c))p)^(2)+(log_((a)/(a))p)^(2))/(((log_((a)/(b))p+log_((b)/(c))p+log_((c)/(a))p))^(2)) wherever defined,simplifies to

Simplify ((p^(2)-10p+21)/(p-7)times(p^(2)+p-12)/((p-3)^(2)))

[P^((b-c))]^(b+c)*[P^((c-a))]^(c+a)*[P^((a-b))]^((a+b))=?

Let 0ltP (A)lt1, 0ltP(B) lt1 and P(A uu B)=P(A)+P(B)-P(A)*P(B) then (A) P(B//A)=P(B)-P(A) (B) P(A^(c)-B^(c))-P(A^(c))-P(B^(c)), (C) P(A uu B)^(c)=P(A^(c))*P(B^(c)) (D) P(A//B)=P(A)

If P(A)=(2)/(3),P(B)=(1)/(2) and P(A cap B)=(1)/(6) , find the value of P(A cup B), P(A//B), P(B//A), P(A cap B^(c )), P(A^(c ) cap B^(c )) and P(A^(c ) cup B^(c ))