Question

\(p\) and \(q\) are positive numbers such that \(p^{q}=q^{p}\), and \(q=9p\). The value of \(p\) is

• A. \(\sqrt{9}\)
• B. \(\sqrt[6]{9}\)
• C. \(\sqrt[9]{9}\)
• D. \(\sqrt[8]{9}\)
• E. \(\sqrt[3]{9}\)

Hint:

For equations of the form \(p^q=q^p\) with \(q=kp\,(k>0)\), the shortcut is \(p=k^1/(k-1)\) obtained by canceling \(p^p\) and taking roots. Ensure \(p>0\) to safely divide by \(p^p\) and take logarithms/roots.

Answer 1

The Correct Option is D

Step 1: Using \(q=9p\) in \(p^{q}=q^{p}\), we get
\[ p^{9p}=(9p)^{p}=9^{p}p^{p}. \]
Divide both sides by \(p^{p}\,(>0)\):
\[ p^{8p}=9^{p}. \]

Step 2: Take the \(p\)-th root (or take logs) to obtain
\[ p^{8}=9 \; \Rightarrow \; p=9^{1/8}=\sqrt[8]{9}. \]
Thus, the required value is \(\boxed{\sqrt[8]{9}}\).