thedude232 wrote:If 2^x - 2^x-2 = 3(2^13), what is the value of x?
a) 9
b) 11
c) 13
d) 15
e) 17
The correct answer is "d". Can someone please show me how to simplify the left side of the equation. Thanks!
2^x -2^(x-2) = 3(2^13) = (2^2-1)(2^13) = 2^15 - 2^13
2^
x - 2^
(x-2) = 2^
15 - 2^
(15-2)
clearly, x = 15 satisfy this equation.
Hence D.
I am adding the NOTE here as just got a private message to explain in a bit detail. Hopefully it clear a bit more now.
{NOTE : 2^x - 2^(x-2) = 3*2^13. As soon as I saw the question I could see, that the right side could be represented in terms of the power of 2.
Lets do it. You can replace 3 by (4-1), in right side:
So, 3*2^13 = (4-1)*2^13 = (2^2 - 1)*2^13 = 2^15 - 2^13
Lets have a look on the equation now :
2^x - 2^(x-2) = 2^15 - 2^13 = 2^15 - 2^(15-2)
Now you can see that if you replace x by 15, it is an identity => x is the solution (No need to solve)
BUT, If you still want to solve the equation here it is :
2^(x-2) [2^2 - 1] = 2^13[2^2 -1]
=> 2^(x-2) = 2^13
=>x-2 = 13
=> x = 15
}
