If 3^x – 3^(x-3) = 2106, what is the value of x? GMAT Factoring Question

If 3^x – 3^(x-3) = 2106, what is the value of x?

This is a fairly tricky factoring question, but it goes back to one of the general factoring principles: always factor out the common term.

Here’s where we start: 

In our case, the common term is 3^x. That leaves us with…

…on the left side of the equation. We’ll deal with the right side later.

This of course simplifies to

The trick at this point will always be to split the fraction:

This can be further factorized as…

This is the simplest that we can get the left side (everything is in Prime Bases), which lends me to think that we’re looking for a factor of 13 in that 2106 on the right side. Let’s factorize 2106 to see: 

Putting this all together, we find:

And of course if we’re solving for x, the only thing that matters is the powers of 3: 

And there you have it: x = 7. 

Let me repeat it here for SEO: If 3^x – 3^(x-3) = 2106, what is the value of x?

If you enjoyed this and would like more examples of factoring questions, check out the Advanced Arithmetic Guide. It’s full of fresh examples of similar factoring questions. Remember, factoring questions are rated as more difficult than they realistically are, so this is an easy way to earn extra points on the GMAT Focus!