GATE 2022 ME (ME2) – Question 8
Consider the following functions for non-zero positive integers, $p$ and $q$. $f(p,q)=p\times p\times\cdots\times p=p^q$ ($q$ terms), $f(p,1)=p$. $g(p,q)$ is the right-associated power tower of $q$ copies of $p$, with $g(p,1)=p$. Which one of the following options is correct based on the above?
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Correct answer: (A) $f(2,2)=g(2,2)$
Explanation
Write out the values with $p,q$ small.
- $f(p,q)=p^q$, so $f(2,2)=4$.
- $g(p,q)$ is the power tower, so $g(2,2)=2^2=4$.
**Option A:** $f(2,2)=4=g(2,2)$. ✓ This is the correct one.
**Checking the others.**
- **B:** $g(2,2)=4$, so $f(g(2,2),2)=f(4,2)=4^2=16$ and $f(2,g(2,2))=f(2,4)=2^4=16$. The two are equal, so "$<$" is false.
- **C:** $g(2,1)=2$ and $f(2,1)=2$, so "$\neq$" is false.
- **D:** $f(3,2)=9$ and $g(3,2)=3^3=27$, so $9>27$ is false.
Only **A** is correct.