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Challenging - Exponent Problem

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by thedude232 » Wed Dec 29, 2010 3:58 pm
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!
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by stormier » Wed Dec 29, 2010 4:09 pm
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) = 2^x-2^x/4 = 2^x(1-1/4)=2^x (3/4) = 3.2^(x-2) =3.2^13 ==> x-2=13 => x = 15
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by anshumishra » Wed Dec 29, 2010 4:09 pm
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
}
Thanks
Anshu

(Every mistake is a lesson learned )
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