Data Sufficiency

2^x - 2^x-2 = 3*(2^13) x=?

a. 9
b. 11
c. 13
d. 15
e. 17

I'm stuck on this one. Any help is much appreciated.

2^(x-2)[2^2 -1] = 3*(2^13) {Taking 2^x-2 common}

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

2^(x-2) * 3 = 3 * (2^13)

Equating powers with same base(2)

x-2 = 13

x = 15

I love this question!

One other method that you can employ on a question like this is to recognize that exponents are extremely pattern-driven. After all, they're just repetitive multiplication (2^13 is 13 2's multiplied together...very, very repetitive math!). So with that in mind, you can test the relationship with small numbers, and if you find a pattern you can extrapolate it to the larger, given numbers.

So here, you'd be assessing the relationship on the left:

2^x - 2^(x-2)

To see what kind of answer you get, and to see if there are patterns as you try different values of x.

Let's try x = 3 (so that x-2 = 1)

2^3 - 2^1 = 8 - 2 = 6

Well, 6 = 3(2), so we do now have something that at least looks like the right-hand side of the equation. Does that 3(2.....) setup always hold? Let's try x = 4:

2^4 - 2^2 = 16 - 4 = 12

And 12 = 3(4) which is also 3(2^2). Now we're getting somewhere...it looks like we'll always get that 3 on the right. So let's try one more to make sure, with x = 5:

2^5 - 2^3 = 32 - 8 = 24

And 24 = 3(8) = 3(2^3)

So it looks like:

-We always get 3(2-to-an-exponent) as our result
-That 2-to-an-exponent takes the form of:

x = 3, x-2 = 1 ---> 3(2^1)
x = 4, x-2 = 2 ---> 3(2^2)
x = 5, x-2 = 3 ---> 3(2^3)

So we always end up with, on the right hand side, 3(2^(x-2)). So to get to 3(2^13), that means that x-2 = 13, so x = 15.


The lessons here:

1) If you don't see to factor here right away (and most won't...the common "x-2" term is tricky), you can test for patterns because exponent questions lend themselves really nicely to patterns.

2) Similarly, if you aren't sure how to get started on a question, trying that same mathematical setup but with easy-to-use numbers helps you to better assess the relationship. Even if you don't go all the way through with patterns, just starting with:

2^3 - 2^1

Lets you know that you can factor out the common second term, so that might help you better see to factor. We're often much more comfortable using small numbers than using either large numbers or variables, so running a "parallel problem" using the same mathematical relationship but smaller numbers can help you to quickly get comfortable with the situation and go from there.