Consider the following functions for non-zero positive integers, $p$ and $q$. 
Which one of the following options is correct based on the above?
To solve the problem, we need to evaluate and compare the functions \(f(p, q)\) and \(g(p, q)\) for the given pairs of positive integers \(p\) and \(q\).
The definitions of the functions are:
- \(f(p, q) = p^q\)
- \(g(p, q) = \underbrace{pppp \ldots}_{q \text{ terms}} = p^{ \text{product of } q \text{ numbers} } = p^q\)
Next, we evaluate the options:
- Option 1: \(f(2,2) = g(2,2)\)
\(f(2, 2) = 2^2 = 4\)
\(g(2, 2) = 2^2 = 4\)
So, \(f(2,2) = g(2,2)\) is correct. - Option 2: \(f(g(2,2), 2) < f(2,g(2,2))\)
\(g(2, 2) = 4\) (as calculated above)
\(f(4, 2) = 4^2 = 16\)
\(f(2, 4) = 2^4 = 16\)
So, \(f(g(2,2), 2) = f(2,g(2,2))\) is incorrect. - Option 3: \(g(2,1) = f(2,1)\)
\(g(2, 1) = 2\) (since it's a product of one term)
\(f(2, 1) = 2\)
So, \(g(2,1) = f(2,1)\) is correct, but not the provided answer option. - Option 4: \(f(3,2) > g(3,2)\)
\(f(3, 2) = 3^2 = 9\)
\(g(3, 2) = 3^2 = 9\)
So, \(f(3,2) = g(3,2)\) is incorrect.
Therefore, the correct answer is indeed Option 1: \(f(2,2) = g(2,2)\).
Important Questions from Powers and Exponents
If a real variable $x$ satisfies $3^{x^2} = 27 \times 9^x$, then the value of $\frac{2^{x^2}}{(2^{x})^2}$ is:
- What is the value of $\left(\frac{3^{81}}{27^4}\right)^{1/3}$?
The 12 musical notes are given as C, C#, D, D#, E, F, F#, G, G#, A, A#. Frequency of each note is $ \sqrt[12]{2} $ times the frequency of the previous note. If the frequency of the note C is 130.8 Hz, then the ratio of frequencies of notes F# and C is:
- For positive integers $p$ and $q$, with $\frac{p}{q} \neq 1$, $(\frac{p}{q})^{\frac{p}{q}} = p^{(\frac{p}{q}-1)}$. Then,
- If $7^{3x} = 216$, the value of $7^{-x}$ (rounded off to three decimal places) is ________.
