If ( 3 2 +i 3 2 )⁵⁰=3²⁵(x+i y) , where x and y are real, then the ordered pair (2 x, 2 y) is

Step-by-step solution

Let z = 3 2 + i 3 2 = 3 ( 3 2 + i 1 2 ) = 3 ( 6 + i 6 ) = 3 e^ i / 6 . Then z⁵⁰ = ( 3 )⁵⁰ e^ i 50 / 6 = 3²⁵ e^ i 25 / 3 . We have e^ i 25 / 3 = e^ i (8 + / 3) = e^ i / 3 = 3 + i 3 = 1 2 + i 3 2 . Given z⁵⁰ = 3²⁵ (x + iy) , we equate the expressions: 3²⁵ (x + iy) = 3²⁵ ( 1 2 + i 3 2 ) . Thus, x = 1 2 and y = 3 2 . The ordered pair (2x, 2y) is (2 1 2 , 2 3 2 ) = (1, 3 ) . Answer: (1, 3 )