Step-by-step explanation:
To solve this expression, let's break it down step by step.
Given expression:
[(p^-1 + q^-1)(p^-1 - q^-1) + (1/(p^-1) - 1/(q^-1))(1/(p^-1) + 1/(q^-1))] * (pq)^2
Step 1: Simplify the terms inside the brackets:
(p^-1 + q^-1)(p^-1 - q^-1) = p^(-1 - 1) - p^(-1)q^(-1) + q^(-1 - 1) - q^(-1) = p^-2 - p^-1q^-1 + q^-2 - q^-1
(1/(p^-1) - 1/(q^-1))(1/(p^-1) + 1/(q^-1)) = (p - q)/(p*q) * (p + q)/(p*q) = (p^2 - q^2)/(p^2*q^2)
Step 2: Substitute the simplified terms back into the expression:
[p^-2 - p^-1q^-1 + q^-2 - q^-1 + (p^2 - q^2)/(p^2*q^2)] * (pq)^2
Step 3: Simplify further:
p^-2 - p^-1q^-1 + q^-2 - q^-1 + (p^2 - q^2)/(p^2*q^2) = (p^2 - 2pq + q^2)/(p^2*q^2) - (pq)/(p*q) + (1/(p*q)) + (p^2 - q^2)/(p^2*q^2)
Step 4: Simplify the expression:
[(p - q)^2 - pq + 1 + (p^2 - q^2)]/(p*q)^2
= [(p^2 - 2pq + q^2) - pq + 1 + p^2 - q^2]/(p*q)^2
= (2p^2 - 2q^2 + 1)/(p*q)^2
So, the simplified answer to the given expression is (2p^2 - 2q^2 + 1)/(p*q)^2. If you have any further questions or need clarification, feel free to ask!
