Question

For positive integers \( p \) and \( q \), with \( \frac{p}{q} \neq 1 \), \[ \left(\frac{p}{q}\right)^{\frac{p}{q}} = \left(\frac{p}{q}\right)^{(\frac{p}{q} - 1)}. \] Then: (1) \( q^p = p^q \)

• A. \( q^p = p^{2q} \)
• B. \( \sqrt{q} = \sqrt{p} \)
• C. \( \sqrt[q]{q} = q\sqrt[p]{p} \)
• D.

Hint:

In exponent-based problems, isolate terms with the same base or power and simplify step by step to uncover hidden relationships.

Answer 1

The Correct Option is A

Step 1: Simplify the given equation.
Starting with:
\[ \left(\frac{p}{q}\right)^q = p^{\frac{p}{q} - 1}. \]
Rewriting:
\[ \frac{p^q}{q^q} = p^{\frac{p}{q} - 1}. \]
Step 2: Identifying the relationship.
Equating powers of \( p \) and \( q \):
\[ p^q = q^p. \]
Step 3: Verification.
The relationship \( p^q = q^p \) satisfies the equation, confirming the solution.