SOLUTION: x = (16 - 8√5)^(1/3) + (16 + 8√5)^(1/3) A = x³ + 12x - 31 Find A²⁰²⁵

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Question 1209541: x = (16 - 8√5)^(1/3) + (16 + 8√5)^(1/3)
A = x³ + 12x - 31
Find A²⁰²⁵
Answer by greenestamps(13199) About Me  (Show Source):
You can put this solution on YOUR website!


If %2816-8sqrt%285%29%29%5E%281%2F3%29 and %2816%2B8sqrt%285%29%29%5E%281%2F3%29 are to be simple expressions, then each of them needs to be of the form a%2Bb%2Asqrt%285%29.



We need to have a%5E3%2B15ab%5E2=16 and 3a%5E2b%2B5b%5E3=8

By inspection, these equations are satisfied when a=b=1. So

%2816-8sqrt%285%29%29%5E%281%2F3%29=1-sqrt%285%29
%2816%2B8sqrt%285%29%29%5E%281%2F3%29=1%2Bsqrt%285%29



Then

A=x%5E3%2B12x-31=2%5E3%2B12%282%29-31=8%2B24-31=1

And finally

A%5E2025=1%5E2025=1

ANSWER: 1