Q:

Solve $3^x+3^{(3x+1)}=108$

I was helping a high school student with their homework and I ran across the following problem: solve $$3^x+3^{(3x+1)}=108.$$ I was unable to find any "elementary way" to do this (by which I mean something a high school student would be comfortable with like manipulating the equation until the bases are the same and then equating the power, or using logarithms to undo exponents). 
Can anyone solve this with $9^{th}$ or $10^{th}$ grade level math?

A:

Follow chubakueno's suggestion of letting $y=3^x$, giving us
$$3y^3+y-108=0$$
The only hope to factor this is for the equation to have a rational root.  According to the rational root theorem, such a root would have to be of the form $\frac pq$ such that $p$ is a factor of $108$ and $q$ is a factor of $3$.  It's also relatively easy to see that there should be one real root between $3$ and $4$.  Since our root is not an integer, we must test $q=3$.  Since neither $10$ nor $11$ are factors of $108$, this root cannot be rational.  High school algebra won't be able to solve this one.

