Question 212297
How do you find {{{x^2}}} if x satisfies 

1. {{{(x+9)^(1/3)-(x-9)^(1/3)=3}}}
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Let {{{A = x+9}}} and {{{B=x-9}}}

2. {{{A^(1/3)-B^(1/3)=3}}}

Cube both sides:

   {{{A-3A^(2/3)B^(1/3)+3A^(1/3)B^(2/3)-B=27}}}

Group the first and last terms on the left together

3. {{{(A-B)-3A^(2/3)B^(1/3)+3A^(1/3)B^(2/3)=27}}}

Notice that {{{A-B=(x+9)-(x-9)=x+9-x+9=18}}}
Substitute {{{18}}} for {{{A-B}}}

{{{18-3A^(2/3)B^(1/3)+3A^(1/3)B^(2/3)=27}}}

Subtract 18 from both sides:

{{{-3A^(2/3)B^(1/3)+3A^(1/3)B^(2/3)=9}}}

Divide through by {{{-3}}}

{{{A^(2/3)B^(1/3)-A^(1/3)B^(2/3)=-3}}}

Factor out {{{A^(1/3)B^(1/3)}}} on the left

{{{A^(1/3)B^(1/3)(A^(1/3)-B^(1/3))=-3}}}

Using equation 2 above, the parenthetical expression
equals 3

{{{A^(1/3)B^(1/3)(3)=-3}}}

Divide both sides by 3

{{{A^(1/3)B^(1/3)=-1}}}

Cube both sides:

{{{(A^(1/3)B^(1/3))^3=(-1)^3}}}

{{{AB=-1}}}

Substitute {{{A = x+9}}} and {{{B=x-9}}}, 

{{{(x+9)(x-9)=-1}}}

{{{x^2-81=-1}}}

Add {{{81}}} to both sides:

{{{x^2=80}}}

That's what you wanted.

Edwin</pre>