It all depends to the quest. since n is an integer and zero is also an integer, so if we substitute zero in place of n we get the least value

Here, n=1, also give the same answer.

Plz see below the properties of Zero,

https://www.rapidtables.com/math/number ... umber.html

As for the ques. yes 1 also provide the same answer but you may be lucky , if the ques states: $(4^n + \frac{(2)}{(4^{n})})$

will n = 0 and n= 1 provide the same answer?

Question states: If n is an integer, what is the least possible value of $(3^n + \frac{(3)}{(3^{n})})$

but you change the basic structure of this question stem which states: $(a^n + \frac{(a)}{(a^{n})})$

You can not change it to 2 in the numerator of second part.
$(4^n + \frac{(2)}{(4^{n})})$


If you follow the stem

if a=3
$(3^n + \frac{(3)}{(3^{n})})$

if a=4
$(4^n + \frac{(4)}{(4^{n})})$

if a=2
$(2^n + \frac{(2)}{(2^{n})})$
........

$(a^n + \frac{(a)}{(a^{n})})$

Always n=0, n=1 works for this because we when n=0, first part is= 1 and second part is a. and
when n=1 first part is= a and second part is 1.

Hence, the result is either 1+a or a+1.

Hope this would clear the confusion about this question.

______________________________________________
Let me know if I am wrong.

For this and all questions it will help you to always consider the core number properties as well as the minimum and the maximum.

The core number properties are 1) Positive/Negative 2) Non-integers and 3) (the special numbers) zero and one

In this case we are told that n is an integer so number 2) non-integers does not apply.

We could think of positive and negative but a negative number would just make 3^-n positive and since it is multiplied by 3 this will result in a larger value.

That leaves us with the third number property, the special numbers -- zero and one. And you do want to plug in both numbers because this question is asking you for the least value so you do need an actual value. It just so happens that zero and one both give you the same answer here.

So it is a combination of logic to tell you which numbers to try and then plugging in those numbers to get an actual value.

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