Problem Solving

kunalkulkarni wrote:2^x - 2^(x-2) = 3(2^13) what is x?

2^(x - 2) = (2^x)*(2^(-2)) = (2^x)/(2^2)

--> (2^x) - 2^(x - 2) = 3*(2^13)
--> (2^x) - (2^x)/(2^2) = 3*(2^13)
--> (2^2)*(2^x) - (2^x) = 3*(2^13)*(2^2)
--> (2^x)*[(2^2) - 1] = 3*(2^15)
--> 3*(2^x) = 3*(2^15)

Comparing the both sides, x = 15

Another tricky method to solve this problem is to expressing the right side of the expression as powers of of 2 only so that we can compare it with the left side.

3*(2^13) = (4 - 1)*(2^13) = (2^2 - 1)(2^13) = 2^15 - 2^13 = 2^15 - 2^(15 - 2)

hence, x = 15

The most important thing when analyzing any GMAT problem is to ask yourself - what concept is being tested? And what is my goal? Always ask yourself these questions before diving in.

Here, we're clearly testing properties of exponents. So what do you know about variable exponents? Well, if you can set the bases equal, you can set the exponents equal. For example, \(3^x = 27\), then \(3^x = 3^3\), so x must equal 3.

In this problem, though, we're subtracting on the left side, and we have a different base on the other side. We know we can only compare the exponents if we express everything as multiplication. So your goal is - How do you get the left side to look like the right side? How do you turn subtraction into multiplication? Well, by factoring! Pull out the biggest common factor from both terms on the left side:

$$2^x-2^{x-2}=3\left(2^{13}\right)$$
$$2^{x-2}\left(2^2-1\right)=3\left(2^{13}\right)$$
$$2^{x-2}\left(4-1\right)=3\left(2^{13}\right)$$
$$2^{x-2}\left(3\right)=3\left(2^{13}\right)$$
$$2^{x-2}=2^{13}$$
$$x-2=13$$
$$x=15$$