Answer: multiplicative inverse of 2 + 4i is:-1/12 (1/(2 - 4i)) = -1/12 (1/(2/10 + 2i/10)) = -1/12 (10/(2^2 + (2i)^2)) = -1/12 (10/(4 + 4)) = -1/12 (10/8) = -5/48 + 0i
Explanation:
The multiplicative inverse of a complex number is found by dividing 1 by the number.The multiplicative inverse of 2 + 4i is given by 1/(2 + 4i).To find this, we can simplify the fraction using basic algebraic rules:1/(2 + 4i) = 1/((2 + 4i)(2 - 4i))/(2 - 4i) = 1/(2^2 + 4^2i^2)/(2 - 4i) = 1/(4 - 16)/(2 - 4i) = 1/(-12)/(2 - 4i) = -1/12/(2 - 4i) = -1/12 (1/(2 - 4i))Since the denominator is in the form of a complex number (2 - 4i), we can find its multiplicative inverse by conjugating the denominator and dividing the numerator by the magnitude squared of the denominator:(2 - 4i)^-1 = (2 + 4i)/((2)^2 + (4i)^2) = (2 + 4i)/(4 + 16) = (2 + 4i)/(20) = (2/20) + (4i/20) = 1/10 + (2i/10)Therefore, the multiplicative inverse of 2 + 4i is:-1/12 (1/(2 - 4i)) = -1/12 (1/(2/10 + 2i/10)) = -1/12 (10/(2^2 + (2i)^2)) = -1/12 (10/(4 + 4)) = -1/12 (10/8) = -5/48 + 0i