If 2^x - 2^x-2 = 3(2^13) then the value of x is?

[guest]

Some someone please help me with the following:

If 2^x - 2^x-2 = 3(2^13) then the value of x is?

The answer is 15 but I cant figure out how to do it.

Thanks

[RonPurewal]

this problem obviously doesn't belong where you originally posted it (in the verbal section), so i will move it to where it _does_ belong.

--

make sure that you know the following:

2 times 2^x = 2^(x+1)
4 times 2^x = 2 times 2 times 2^x = 2^(x+2)
etc.

so, the left side simplifies to
(2^x) - (2^(x-2))
= (2 times 2 times 2^(x-2)) - (2^(x-2))
= 4(2^(x-2)) - (2^(x-2))
= 3(2^(x-2))

...and there you go.

[fagnerdonadon]

I'm really sorry but I just can't understand what you did... I understood the first part but when you apply the rule to the exercise itself I don't understand.... please help. Thanks.

[joshua.higgins]

so, the left side simplifies to
(2^x) - (2^(x-2))
= (2 times 2 times 2^(x-2)) - (2^(x-2))
= 4(2^(x-2)) - (2^(x-2))
= 3(2^(x-2))

You factor out the 2^(x-2), so you get:

= 4(2^(x-2)) - (2^(x-2))
=2^(x-2)(4*1 - 1)
=3(2^(x-2))
At that point you can drop the 3's from each side, so the equation becomes:
2^(x-2)=2^13
Once you have the same bases:
x-2=13
x=15

[RonPurewal]

try the following other thread on this problem:

post16795.html#p16795

[johnhillescobar]

I'd like to give you my approach:

First, you have to realize that 2^-2 = 1/2^2. Therefore 2^(x-2) = 2^x/2^2. So:

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

2^x(1-1/2^2)=3(2^13) factor 2^x from the first term

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

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

2^x=4(2^13) multiply both sides by 3/4 reciprocal

2^x=2^2(2^13)

2^x=(2^15)

So x = 15

[Ben Ku]

johnhillescobar, your solution is great! There are usually many good solutions to math problems.

[gol10dr1]

First, you have to realize that 2^-2 = 1/2^2. Therefore 2^(x-2) = 2^x/2^2. So:

2^x(1-1/2^2)=3(2^13) factor 2^x from the first term

John, those two parts make absolutely no sense to me. Could you please explain how you got 2^(x-2)=2^x/2^2? I'm completely lost. Thanks

[mara.rajesh]

3(2^13) can be written as (4-1)(2^13) so

2^x - 2^x-2=(4-1)(2^13)

2^x - 2^x-2=(2^2 -2^0)(2^13)

2^x - 2^x-2=(2^2*2^13)-(2^0*2^13)

When bases are equal exponents are added so

2^x - 2^x-2=2^15-2^13

which concludes x value as 15.

[mschwrtz]

Nice mara.rajesh.

In answer to your question, gol10dr1:

One rule of exponents states that (n^a)/(n^b)=n^(a-b).
John applies that rule here to get (2^x)/(2^2)=2^(x-2).

[azarmosk]

Hi mara.rajesh...

Your explanation is pretty clear however, I dont understand after below equation.. how does this lead to 15?



2^x - 2^x-2=2^15-2^13

which concludes x value as 15.

[adiagr]

Hi mara.rajesh...

Your explanation is pretty clear however, I dont understand after below equation.. how does this lead to 15?



2^x - 2^x-2=2^15-2^13

which concludes x value as 15.

Hi Azarmosk,

Just compare exponents in the Left hand side and Right Hand side of equation


2^x - 2^x-2=2^15-2^13

Thus
2^x =2^15

From where we can see that x is 15.

Cross check:


2^x-2=2^13


so x-2 =13, from where again x comes out as 15.

I hope it is clear now.

Aditya

[RonPurewal]

Hi mara.rajesh...

Your explanation is pretty clear however, I dont understand after below equation.. how does this lead to 15?



2^x - 2^x-2=2^15-2^13

which concludes x value as 15.

Hi Azarmosk,

Just compare exponents in the Left hand side and Right Hand side of equation


2^x - 2^x-2=2^15-2^13

Thus
2^x =2^15

From where we can see that x is 15.

Cross check:


2^x-2=2^13


so x-2 =13, from where again x comes out as 15.

I hope it is clear now.

Aditya

yep.