Question 1205857
.
Find x,
(7 + √x)^⅓ + (7 - √x)^⅓ = 2
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<pre>

Your starting equation is

    {{{(7+sqrt(x))^(1/3)}}} + {{{(7-sqrt(x))^(1/3)}}} = 2.    (1)


Let  a = {{{(7+sqrt(x))^(1/3)}}};  b = {{{(7-sqrt(x))^(1/3)}}}.


Raise both sides of equation (1) to degree 3.

Use

    {{{(a + b)^3}}} = {{{a^3 + b^3}}} + 3ab*(a+b),


    {{{a^3 + b^3}}} = {{{(7+sqrt(x))}}}  + {{{(7-sqrt(x))}}} = 14,

    a + b = 2   <<<---===  given by equation,

    ab = {{{(49 - x)^(1/3)}}}.


You will get

    {{{14 + 3*(49-x)^(1/3)*2}}} = 8.


Simplify and find x

    {{{(49 - x)^(1/3)}}} = {{{(8-14)/6}}},

    {{{(49 - x)^(1/3)}}} = -1.


Raise both sides of the last equation to degree 3

    49 - x = -1,

    49 + 1 = x,

     x     = 50.


<U>ANSWER</U>.  x = 50.


<U>CHECK</U>.   {{{(7 + sqrt(50))^(1/3)}}} + {{{(7 - sqrt(50))^(1/3)}}} = 2.414213562 + (-0.414213562) = 2.0000000
</pre>

Solved.