Question

For positive integers $p$ and $q$, with $\tfrac{p}{q} \neq 1$, $\Big(\tfrac{p}{q}\Big)^{\tfrac{p}{q}} = p^{\big(\tfrac{p}{q}-1\big)}$. Then,

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

Hint:

When dealing with equations involving fractional exponents, simplify step by step and carefully use laws of indices. Multiplying across and canceling terms is often the key.

Answer 1

The Correct Option is A

Step 1: Start with the given equation. \[ \left(\frac{p}{q}\right)^{\tfrac{p}{q}} = p^{\left(\tfrac{p}{q}-1\right)} \] Step 2: Rewrite the RHS. \[ p^{\left(\tfrac{p}{q}-1\right)} = p^{\tfrac{p}{q}} \cdot p^{-1} = \frac{p^{\tfrac{p}{q}}}{p} \] Step 3: Compare both sides. \[ \left(\frac{p}{q}\right)^{\tfrac{p}{q}} = \frac{p^{\tfrac{p}{q}}}{p} \] Multiply both sides by $p$: \[ p\left(\frac{p}{q}\right)^{\tfrac{p}{q}} = p^{\tfrac{p}{q}} \] Step 4: Simplify the LHS. \[ p \cdot \frac{p^{\tfrac{p}{q}}}{q^{\tfrac{p}{q}}} = p^{\tfrac{p}{q}} \] Cancel $p^{\tfrac{p}{q}}$ (non-zero): \[ \frac{p}{q^{\tfrac{p}{q}}} = 1 \] Step 5: Rearrange. \[ q^{\tfrac{p}{q}} = p \] Raise both sides to power $q$: \[ q^p = p^q \] Thus, \[ \boxed{q^p = p^q} \]