To find the value of the given expression, we start by analyzing the provided equation involving positive integers p and q. The problem presents an initial equation: \(\frac{p}{q}+\frac{q}{p}=3\). Our objective is to determine the value of a related expression: \(\frac{p^2}{q^2} + \frac{q^2}{p^2}\).

Expression Identity and Approach

This type of mathematical problem frequently involves recognizing and applying algebraic identities. We are given a sum of two terms, \(\frac{p}{q}\) and \(\frac{q}{p}\), and we need to find the sum of their squares. A fundamental algebraic identity that connects these forms is the square of a sum: \((a+b)^2 = a^2+2ab+b^2\).

Let's simplify the problem by assigning variables to the terms in our given equation. If we let \(a = \frac{p}{q}\) and \(b = \frac{q}{p}\), then the initial equation can be written as \(a+b=3\).

The algebraic expression we need to find is \(a^2+b^2\).

p and q Expression Derivation

We can use the algebraic identity \((a+b)^2 = a^2+2ab+b^2\) to establish a relationship between the given equation and the required expression. By rearranging this identity, we can isolate \(a^2+b^2\):

\(a^2+b^2 = (a+b)^2 - 2ab\)

Now, substitute \(a = \frac{p}{q}\) and \(b = \frac{q}{p}\) back into this rearranged algebraic identity:

\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = \left(\frac{p}{q} + \frac{q}{p}\right)^2 - 2 \left(\frac{p}{q}\right) \left(\frac{q}{p}\right)\)

Observe the product term \(\left(\frac{p}{q}\right) \left(\frac{q}{p}\right)\). This simplifies nicely:

\(\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = \frac{p \cdot q}{q \cdot p} = 1\)

Substituting this simplification back into our expression for the sum of squares:

\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = \left(\frac{p}{q} + \frac{q}{p}\right)^2 - 2(1)\)

\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = \left(\frac{p}{q} + \frac{q}{p}\right)^2 - 2\)

We are given in the problem statement that \(\frac{p}{q}+\frac{q}{p}=3\). Now, we substitute this value into the simplified expression:

\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = (3)^2 - 2\)

\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = 9 - 2\)

\(\frac{p^2}{q^2} + \frac{q^2}{p^2} = 7\)

Final Expression Value

The calculated value for the expression \(\frac{p^2}{q^2} + \frac{q^2}{p^2}\) is 7.