p and q are positive integers and $\frac{p}{q}+\frac{q}{p}=3$

then $\frac{p^2}{q^2}+\frac{q^2}{p^2}=$

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.