Question

For positive integers \( p \) and \( q \), with \( \frac{p}{q} \neq 1 \), \( \left( \frac{p}{q} \right)^q = p^{q-1 \). Then, which of the following is true?}

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

Hint:

When dealing with powers and exponents, remember to manipulate the terms carefully to isolate variables and solve for them systematically.

Answer 1

The Correct Option is D

We are given the equation: \[ \left( \frac{p}{q} \right)^q = p^{q-1} \] \[ \Rightarrow \frac{p^q}{q^q} = p^{q-1} \] Multiply both sides by \( q^q \) to clear the denominator: \[ p^q = q^q \cdot p^{q-1} \] Now divide both sides by \( p^{q-1} \) (assuming \( p \neq 0 \)): \[ p = q^q \] This implies that: \[ p^q = q^p \] Thus, the correct answer is option (4).