I have another GMAT question. I am able to get it right, but just out of logic :(. Bad logic.?

Question

equation is: 2^x- 2^(x-2)= 3(2^13). Solve for X

^ preceeds exponents. Thank you!

Answer 1

Remember this rule of exponants?

a^x * a^y = a^(x + y)

Use this rule to re-write your equation as

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

This is the same as

2^x - ( 2^x * (1/4) ) = 3(2^13) = 24576

or

2^x - (2^x)/4 = 3(2^13)

don't get bothered by those 2^x terms you have one of them and you are going to subtract 1/4 of one from it, leaving you with 3/4 of one

(3/4) * (2^x) = 3(2^13)

multiply both sides by (4/3)

2^x = 4(2^13)

since 4 = 2^2 you have

2^x = 2^2 * 2^13

or

2^x = 2^15

so x = 15

Answer 2

Factor the left hand side.

2^x - 2^(x-2) = 3(2^13)
2^(x-2) [2² - 1] = 3(2^13)
2^(x-2) [4 - 1] = 3(2^13)
3[2^(x-2)] = 3(2^13)

2^(x-2) = 2^13

x - 2 = 13
x = 15

Answer 3

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

so X=15

Answer 4

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