Question 1046716
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Kindly help me solving it
By solving equation {{{(1+x)^(2/3)+(1-x)^(2/3)=4 (1-x^2)^(1/3)}}} the values of x are?
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{{{(1+x)^(2/3)+(1-x)^(2/3)}}} = {{{4*(1-x^2)^(1/3)}}}.     (1)

Let us introduce new variables  a = {{{(1+x)^(1/3)}}},  b = {{{(1-x)^(1/3)}}}.  Then the equation (1) takes the form

{{{a^2 + b^2}}} = {{{4ab}}}.                           (2)

Divide both sides of (2) by {{{b^2}}}. You will get

{{{(a/b)^2 + 1}}} = {{{4*(a/b)}}}.                        (3)

Let us introduce new variable z = {{{a/b}}}.  Then the equation (3) takes the form

{{{z^2 + 1}}} = 4z,   or

{{{z^2 -4z + 1}}} = {{{0}}}.

Solve it by using the quadratic formula. You will get

{{{z[1,2]}}} = {{{(4 +- sqrt(16-4))/2}}} = {{{2 +- sqrt(3)}}}.

Now you come to the equations  for x:

{{{(1+x)/(1-x)}}} = {{{(2 + sqrt(3))^3}}}  and/or  

{{{(1+x)/(1-x)}}} = {{{(2 - sqrt(3))^3}}}.

It is just a technique to solve them.

I think it will not be difficult for the person who comes with such a request.
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