SOLUTION: Hi, I really really need to submit my math homeworks as soon as possible, but I cant solve this one question. I need your help. Thank you so much in advance. :) 2^(2x) - 2^(x+2

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Hi, I really really need to submit my math homeworks as soon as possible, but I cant solve this one question. I need your help. Thank you so much in advance. :) 2^(2x) - 2^(x+2      Log On


   

Question 1002287: Hi,
I really really need to submit my math homeworks as soon as possible, but I cant solve this one question. I need your help. Thank you so much in advance. :)
2^(2x) - 2^(x+2) = 32
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
2%5E%282x%29+-+2%5E%28x%2B2%29+=+32

Use the laws of exponents 
        %28a%5Eb%29%5Ec=a%5E%28bc%29 and a%5Eb%2Aa%5Ec=a%5E%28b%2Bc%29
in reverse as
        a%5E%28bc%29=%28a%5Eb%29%5Ec and a%5E%28b%2Bc%29=a%5Eb%2Aa%5Ec

to rewrite the terms on the left of

2%5E%282x%29+-+2%5E%28x%2B2%29+=+32

as

%282%5Ex%29%5E2+-+2%5Ex%2A2%5E2+=+32

%282%5Ex%29%5E2+-+2%5Ex%2A4+=+32

%282%5Ex%29%5E2+-+4%282%5Ex%29+=+32

%282%5Ex%29%5E2+-+4%282%5Ex%29-+32=0

Let u = 2x

u%5E2-4u-32=0

Factor the left side:

%28u-8%29%28u%2B4%29=0

u-8 = 0;   u+4 = 0
  u = 8;     u = -4

Since u = 2x

2x = 8;  2x = -4

The second has no real solution so we ignore it.
 
2x = 8

2x = 23

The bases are equal and positive and not 1, so 
the exponents are equal.

x = 3

Edwin