Roots are in the form of p:q, then find √p/√q + √q/√p

Given that the roots are in the form \( p:q \), we need to find the expression \( \frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}} \).

### Step 1: Simplify the expression
We can start by simplifying the expression:

\[
\frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}}
\]

This can be rewritten using the property \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \):

\[
\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}}
\]

### Step 2: Further simplify
The expression \( \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} \) can be combined as follows:

\[
\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} = \sqrt{\frac{p}{q}} + \frac{1}{\sqrt{\frac{p}{q}}}
\]

### Step 3: Introduce a new variable
Let \( x = \sqrt{\frac{p}{q}} \). Then the expression becomes:

\[
x + \frac{1}{x}
\]

### Step 4: Simplify using algebraic identity
The expression \( x + \frac{1}{x} \) can be interpreted as:

\[
x + \frac{1}{x} = \frac{x^2 + 1}{x}
\]

But in most cases, \( x + \frac{1}{x} \) is left as it is because the result depends on the specific values of \( p \) and \( q \).

### Final Answer:
So, the expression \( \frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}} \) simplifies to:

\[
\boxed{x + \frac{1}{x}} \quad \text{where} \quad x = \sqrt{\frac{p}{q}}
\]

This result is symmetric and shows the relationship between \( p \) and \( q \) in the ratio \( p:q \).

Would you like more details on this, or do you have any other questions?

### Related Questions:
1. How do you find the value of \( x + \frac{1}{x} \) for specific values of \( p \) and \( q \)?
2. What is the significance of the expression \( x + \frac{1}{x} \) in other mathematical contexts?
3. How does the ratio \( p:q \) affect the properties of the expression \( \frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}} \)?
4. How can you derive the expression \( \frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}} \) using different methods?
5. What are the applications of the expression \( \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} \) in algebra or geometry?

### Tip:
Whenever dealing with square roots and ratios, remember to explore both direct simplifications and introducing variables to simplify complex expressions.

Explore how to simplify the expression √p/√q + √q/√p using algebraic techniques. This problem involves understanding the relationship between square roots in a ratio form and their reciprocal sums. Suitable for advanced high school students and anyone interested in algebraic manipulations.

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