Help to solve binominal theorem

[pusekatja]

Hi!
I am struggling to solve this question about binomial theorem, could someone help me? :)

Consider the expansion of $$x^2\left(3x^2+\frac{k}{x}\right)^8$$
The constant term is 16128.
Find k.

 

[MarkFL]

Hello and welcome to MHB, pusekatja! :D

Without knowing what you have tried, I can't know where you are stuck, so this is why we ask our users to show what they have tried so far.

Now, according to the binomial theorem, the $i$th term of the given expansion will be:

$$x^2{8 \choose i}\left(3x^2\right)^{8-i}\left(\frac{k}{x}\right)^i$$

How about we rewrite this term as follows:

$$3^{8-i}{8 \choose i}k^ix^{18-3i}$$

Now, what does this tell us $i$ must be in order for the term to be a constant?