The value of ( +i )^4 ( +i )^5 = where i= -1

Step-by-step solution

To evaluate the expression ( +i )^4 ( +i )^5 , we apply De Moivre's Theorem and properties of complex numbers in polar form. Using De Moivre's Theorem, the numerator simplifies to (4 ) + i (4 ) . For the denominator, we first express + i in polar form as ( 2 - ) + i ( 2 - ) . Applying De Moivre's Theorem gives ( +i )^5 = ( 5 2 -5 ) + i ( 5 2 -5 ) . Dividing these complex numbers yields (9 - 5 2 ) + i (9 - 5 2 ) . Using the periodicity of trigonometric functions, this simplifies to (9 - 2 ) + i (9 - 2 ) . Applying t