<PRE><B>Question 1209541</B>
&lt;br&gt;
If {{{(16-8sqrt(5))^(1/3)}}} and {{{(16+8sqrt(5))^(1/3)}}} are to be simple expressions, then each of them needs to be of the form {{{a+b*sqrt(5)}}}.&lt;br&gt;
{{{(a+b*sqrt(5)^3)=a^3+3a^2b*sqrt(5)+3a(5b^2)+5b^3sqrt(5)=(a^3+15ab^2)+(3a^2b+5b^3)*sqrt(5)}}}&lt;br&gt;
We need to have {{{a^3+15ab^2=16}}} and {{{3a^2b+5b^3=8}}}&lt;br&gt;
By inspection, these equations are satisfied when a=b=1. So&lt;br&gt;
{{{(16-8sqrt(5))^(1/3)=1-sqrt(5)}}}
{{{(16+8sqrt(5))^(1/3)=1+sqrt(5)}}}&lt;br&gt;
{{{(16-8sqrt(5))^(1/3)+(16+8sqrt(5))^(1/3)=(1-sqrt(5))+(1+sqrt(5))=2}}}&lt;br&gt;
Then&lt;br&gt;
{{{A=x^3+12x-31=2^3+12(2)-31=8+24-31=1}}}&lt;br&gt;
And finally&lt;br&gt;
{{{A^2025=1^2025=1}}}&lt;br&gt;
ANSWER: 1&lt;br&gt;

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