Let be a solution of x^2+x+1=0 , and for some a and b in R , [ array lll 4 & a & b array ] [ array ccc 1 & 16 & 13 -1 & -1 & 2 -2 & -14 & -8 array ]= [ array ccc 0 & 0 & 0 array ] . If 4 ^4 + m ^ a + n ^ b =3 , then m + n is equal to

Step-by-step solution

x^2+x+1=0 is root aligned & ^2+ +1=0 & = as ^2 [cube root of unity] aligned also . array l [ array ll 4-a-2 b & 64-a-14 b array 52+2 a-8 b . array ] aligned & = [ array ll 0 & 0&0 array ] & a+2 b=4 & a+14 b=64 & 12 b=60 b=5 & a=-6 & 4 ^4 + m ⁻⁶ + n ^5 =3 & 4 + m 1 + n ^2 =3 & 4 ^2+m+n =3 aligned aligned & 4 (- 1 2 - 3 2 i )+ m + n (- 1 2 + 3 2 i )=3 & -2+ m - n 2 =3....(1) & & -4 3 2 + n 3 2 =0 & n =4 & ~m =7 & ~m + n =11 aligned