SOLUTION: Could you please help me with how to work out this question: By letting {{{ y = x^(1/3) }}} or otherwise, find the values of x for which {{{ x^(1/3) - 2x^(-1/3) = 1 }}} . I h

Algebra ->  Exponents-negative-and-fractional -> SOLUTION: Could you please help me with how to work out this question: By letting {{{ y = x^(1/3) }}} or otherwise, find the values of x for which {{{ x^(1/3) - 2x^(-1/3) = 1 }}} . I h      Log On


   

Question 647659: Could you please help me with how to work out this question:
By letting +y+=+x%5E%281%2F3%29+ or otherwise, find the values of x for which +x%5E%281%2F3%29+-+2x%5E%28-1%2F3%29+=+1+ .
I have the answers (-1 and 8) in the back of my textbook but I am unsure of how to deal with negative fractional indices in this case, and I have to show my working out in my answer so please could you show me how to work this out!
Thank you.
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
ok let assume

+y+=+x%5E%281%2F3%29+.....1
then
+x%5E%281%2F3%29+-+2x%5E%28-1%2F3%29+=+1+ .....1
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+x%5E%281%2F3%29+-+2x%5E%28-1%2F3%29+=+1+ .....1 ...solve for x..
note, because a%5E%28-n%29+means 1%2Fa%5En we will have
+1%2Fx%5E3+-+2%281%2Fx%5E3%29+=+1+ .......then replace x%5E1%2F3 with y
then you get

+y+-+2%281%2Fy%29+=+1+.........simplify
+%28y%5E2+-+2%29%2Fy+=+1+
+y%5E2+-+2+=+y+
+y%5E2+-y-+2+=+0+...from this we can get
%28y-2%29+%28y%2B1%29+=+0
we can get value of y from y-2+=+0 then y+=+2
and from y%2B1+=+0 then y+=+-1
continue to put the value of y to x
y+=+x%5E1%2F3
assume y+=+2
x%5E1%2F3+=+2 -> x+=+2%5E3 -> highlight%28x+=+8%29
assume y+=+-1
x%5E1%2F3+=+-1 -> x+=+-1%5E3 -> highlight%28x+=+-1%29