We start off with what's given:
((2p^3q^2)/(8p^4q))/((4pq^2)/(16p^4))2p3q28p4q4pq216p4
Recall: (a/b)/(c/d)=(a/b)/(c/d)*(d/c)/(d/c)=a/b*d/cabcd=abcd⋅dcdc=ab⋅dc
So,
((2p^3q^2)/(8p^4q))/((4pq^2)/(16p^4))=((2p^3q^2)/(8p^4q))*((16p^4)/(4pq^2))2p3q28p4q4pq216p4=(2p3q28p4q)⋅(16p44pq2)
Recall: a^5/a^3=a^5*a^-3=a^(5-3)=a^2a5a3=a5⋅a−3=a5−3=a2
So,
((2p^3q^2)/(8p^4q))((16p^4)/(4pq^2))(2p3q28p4q)(16p44pq2)==((p^3q^2)/(4p^4q))((4p^4)/(pq^2))(p3q24p4q)(4p4pq2)
The ((p^3q^2)/(4p^4q))(p3q24p4q) part becomes:
((p^3q^2)/(4p^4q))=(p^3q^2)(4^(-1)p^(-4)q^(-1))=4^(-1)p^(3-4)q^(2-1)=4^(-1)p^(-1)q^1(p3q24p4q)=(p3q2)(4−1p−4q−1)=4−1p3−4q2−1=4−1p−1q1
=q/(4p)=q4p
The ((4p^4)/(pq^2))(4p4pq2) part becomes:
((4p^4)/(pq^2))=(4p^4)(p^(-1)q^(-2))=4p^(4-1)q^(-2)=4p^3q^(-2)(4p4pq2)=(4p4)(p−1q−2)=4p4−1q−2=4p3q−2
=(4p^3)/q^2=4p3q2
So, our original setup was:
((2p^3q^2)/(8p^4q))/((4pq^2)/(16p^4))=((2p^3q^2)/(8p^4q))*((16p^4)/(4pq^2))=((p^3q^2)/(4p^4q))((4p^4)/(pq^2))2p3q28p4q4pq216p4=(2p3q28p4q)⋅(16p44pq2)=(p3q24p4q)(4p4pq2)
=(q/(4p))((4p^3)/q^2)=(4p^3q)/(4pq^2)=p^2/q=(q4p)(4p3q2)=4p3q4pq2=p2q