Question

For positive integers \( p \) and \( q \), with \( \frac{p}{q} \neq 1 \), \[ \left(\frac{p}{q}\right)^{\frac{p}{q}} = \left(\frac{p}{q}\right)^{(p - q)}. \] Then: (1) \( q^p = p^q \)
(2) \( q^p = p^{2q} \)
(3) \( \sqrt{q} = \sqrt{p} \)
(4) \( \sqrt[q]{q} = q\sqrt[p]{p} \)

• A.
• B.
• C.
• D.

Hint:

For problems involving exponents, simplify step-by-step, focusing on separating the variables and their respective powers.

Answer 1

The Correct Option is A

Step 1: Start with the given equation. The given equation is: \[ \left( \frac{p}{q} \right)^q = p^{\left(\frac{p}{q} - 1\right)}. \] Rewriting \( \left( \frac{p}{q} \right)^q \): \[ \left( \frac{p}{q} \right)^q = \frac{p^q}{q^q}. \] Thus, \[ \frac{p^q}{q^q} = p^{\left(\frac{p}{q} - 1\right)}. \] Step 2: Analyze the powers of \( p \) and \( q \). From the equation above, equate the terms involving \( p \) and \( q \): \[ p^q = q^p. \] Step 3: Verifying the result. The relationship \( p^q = q^p \) satisfies the given equation and matches option (a).