Given x + 1/x = 1

=> (x² + 1) = x

=> x² - x + 1 = 0

=> x = {-(-1) ± √(1² - 4*1*1)}/(2*1)

=> x = {1 ± √(1 - 4)}/2

=> x = {1 ± √(-3)}/2

=> x = {1 ± √(-1)*√3}/2

=> x = {1 ± i√3}/2                   {since i = √(-1)}

=> x = -w, -w²

Now, put x = -w, we get

         x²⁰⁰⁰ + 1/x²⁰⁰⁰ = (-w)²⁰⁰⁰ + 1/(-w )²⁰⁰⁰

                                   = w²⁰⁰⁰ + 1/w²⁰⁰⁰

                                   = w²⁰⁰⁰ + 1/w²⁰⁰⁰

                                   = {(w³ )⁶⁶⁶ * w² } + 1/{(w³ )⁶⁶⁶ * w² }

                                   = w² + 1/w²                                                 {since w³ = 1}

                                   = w² + w³ /w²

                                   = w² + w

                                   = -1                            {since 1 + w + w² = 0}

So, x²⁰⁰⁰ + 1/x²⁰⁰⁰ = -1