SOLUTION: I really need help with this equation with rational exponents. This problem is not from a text book, it is from a worksheet. (x-5)raised to the two-third power +3(x-5)raised to t

Algebra ->  Algebra  -> Rational-functions -> SOLUTION: I really need help with this equation with rational exponents. This problem is not from a text book, it is from a worksheet. (x-5)raised to the two-third power +3(x-5)raised to t      Log On

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Question 179673: I really need help with this equation with rational exponents. This problem is not from a text book, it is from a worksheet.
(x-5)raised to the two-third power +3(x-5)raised to the one-third power-4=0 ( I couldn't figure out how to write this problem using the formula plotting system.)
I have no idea how to work this problem. Please help!!!
Answer by Earlsdon(6103) About Me  (Show Source):
You can put this solution on YOUR website!
You can temporarily chang variables to facilitate solving problems like this:
%28x-5%29%5E%282%2F3%29+%2B+3%28x-5%29%5E%281%2F3%29-4+=+0 You can rearrange this as:
%28%28x-5%29%5E%281%2F3%29%29%5E2%2B3%28%28x-5%29%5E%281%2F3%29%29-4+=+0
Let's substitute y+=+%28x-5%29%5E%281%2F3%29 to give us:
y%5E2%2B3y-4+=+0 Factor this:
%28y-1%29%28y%2B4%29+=+0 from which:
y+=+1 or y+=+-4 Replacing the variable: y+=+%28x-5%29%5E%281%2F3%29 we get:
%28x-5%29%5E%281%2F3%29+=+1 or %28x-5%29%5E%281%2F3%29+=+-4 so...
If %28x-5%29%5E%281%2F3%29+=+1 Raise each side to the third power.
x-5+=+1%5E3 but 1%5E3+=+1 so...
x-5+=+1 Add 5 to both sides.
highlight%28x+=+6%29 and...
%28x-5%29%5E%281%2F3%29+=+-4 Cube both sides.
x-5+=+%28-4%29%5E3
x-5+=+-64 Add 5 to both sides.
highlight%28x+=+-59%29