Algebraic Power Question

Here’s a question that I got stumped on during the test. Give me enough time (15 minutes), and I’ll figure it out.

Given 2^x – 2^x-2 = 3 (2^13), what is the value of x?

There are two ways of going about this. The first approach is to solve for x. Below are the steps.

[1] 2^x – (1/4)2^x = 3(2^13)
[2] 2^x(1 – 1/4) = 3(2^13)
[3] 2^x(3/4) = 3(2^13)
[4] (1/3)(3/4)2^x = 2^13
[5] (1/4)2^x = 2^13
[6] 2^x-2 = 2^13
[7] x-2 = 13
[8] x = 15

Solving for x algebraiclly is long and error prone (at least for me). My problem was I didn’t know how to deal with the 2^x-2 part of the equation. This led to me to the second approach: back solving.

Back solving involves plugging in numbers from the answers and doing so until you find the correct answer. The recommended strategy is to start from E and work your way back to A. Answer E happens to be 15, so back solving with E gives you:

2^15 – 2^15-2 = 3(2^13)
2^15 – 2^13 = 3(2^13)
(1/2^13)(2^15-2^13) = 3
2^2 – 1 = 3
4 – 1 = 3
3 = 3

At first glance, back solving looks like it will take just about as many steps as solving for x straight up.  Back solving though is a good strategy to use if you don’t know the “right” way of solving a problem. For this question, back solving isn’t as bad as it looks. The calculations are a lot simpler. Just don’t make a silly mistake like I did and choose the wrong answer after calculating the right one