If and are the distinct roots of equation x^2-x+1=0 , then what is the value of | ¹⁰⁰+ ¹⁰⁰ ¹⁰⁰- ¹⁰⁰ | ?

Step-by-step solution

x^2-x+1=0 We know that roots of x^2+x+1=0 are and ^2 . Roots of x^2-x+1=0 are - and - ^2 where , ^2 are complex cube roots of unity | ¹⁰⁰+ ¹⁰⁰ ¹⁰⁰- ¹⁰⁰ |= | ¹⁰⁰+ ( ^2 )¹⁰⁰ ¹⁰⁰- ( ^2 )¹⁰⁰ |= | + ^2 - ^2 |= | -1 - ^2 |= | -1 -1+ 3 i 2 - (-1- 3 i) 2 |= | -1 3 i |= | i 3 |= 1 3