SOLUTION: Solve for x: 2(8)^(4x)-33(8)^(2x)+16=0

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Question 135305: Solve for x: 2(8)^(4x)-33(8)^(2x)+16=0
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Solve for x:
2%288%29%5E%284x%29-33%288%29%5E%282x%29%2B16+=+0
Let's temporarily substitute z+=+%288%29%5E%282x%29 and z%5E2+=+%288%29%5E%284x%29
2z%5E2-33z%2B16+=+0 This can be solved for z by factoring.
%282z-1%29%28z-16%29+=+0 and...
2z-1+=+0 and z-16+=+0, so we have...
2z+=+1 so that:
z+=+1%2F2 and
z+=+16
Now we replace the z with z+=+%288%29%5E%282x%29, so...
%288%29%5E%282x%29+=+1%2F2 but 8+=+2%5E3 and 1%2F2+=+2%5E%28-1%29, so, making these substitutions, we get:
%282%5E3%29%5E%282x%29+=+%282%29%5E%28-1%29 Simplify the left side.
%282%29%5E%286x%29+=+%282%29%5E%28-1%29 Applying the rule: Ifm%5Ea+=+m%5Eb then a+=+b, so...
6x+=+-1 Divide both sides by 6.
x+=+-1%2F6 This is one of the roots.
z+=+16 Substitute z+=+%288%29%5E%282x%29
%288%29%5E%282x%29+=+16 Substitute: 8+=+2%5E3 and 16+=+2%5E4
%282%5E3%29%5E%282x%29+=+%282%29%5E4 Simplify the left side.
%282%29%5E%286x%29+=+%282%29%5E4 or...
6x+=+4 Divide both sides by 6.
x+=+4%2F6 Simplify.
x+=+2%2F3 This is the second root.
Solution is:
x+=+-1%2F6
x+=+2%2F3
I'll leave the check for you to do. It's not difficult and I've done it myself and the two solutions for x are indeed valid!