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a^3 +1/a^3=? and a^5 +1/a^5 =? if (a+1/a)^2=11

Source: 15th -a QEDMO problem 4 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

May 29, 2021
algebra

Problem Statement

Let aa be a real number such that (a+1a)2=11\left(a + \frac{1}{a}\right)^2=11. What possible values can a3+1a3a^3 + \frac{1}{a^3} and a5+1a5a^5 + \frac{1}{a^5} take?
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