Useful property: \(\frac{x^n}{y^n} = (\frac{x}{y})^n\)

We can solve this question using matching operations

Given:
Quantity A: \(\frac{1}{3^n}\)

Quantity B: \(3 (\frac{1}{4^n})\)


Multiply both quantities by \(4^n\) to get:
Quantity A: \(\frac{4^n}{3^n}\)

Quantity B: \(3\)


Apply the property above to get:
Quantity A: \((\frac{4}{3})^n\)

Quantity B: \(3\)

At this point we might readily recognize that by changing the value of n, we can vary the value of Quantity A.
We can demonstrate this by testing some different values of n.

case i: Since n is a positive integer it COULD be the case that n = 1
We get:
Quantity A: \((\frac{4}{3})^1 = \frac{4}{3}\)
Quantity B: \(3\)
In this case, Quantity B is greater


case ii: Since n is a positive integer it COULD be the case that n = 4
We get:
Quantity A: \((\frac{4}{3})^4 = \frac{256}{81}\) = some number greater than 3
Quantity B: \(3\)
In this case, Quantity A is greater

Answer: D

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