# De Moivre’s Theorem – Complex Numbers – Engineering Mathematics 1

Hi Friends, Now I’ll ask you one question if you want to find out square root or let’s say cube root or the fourth root of a complex number then how you will find now as we find roots or the powers in real number you can still find roots or powers or certain expansions of complex number also but how to find it so in this topic I am gonna to teach you how to find certain expansions roots and powers of complex numbers now this now this is this has become possible only because of one scientist whose name is Abraham D-moyos so he has derived a formula using which we can very easily find out roots or the powers or expansion of complex numbers as we do with real numbers so that is what this theorem is called as d-boy was theorem so name of the topic that we are starting today is nothing but demo was theorem now what abraham de Maire says so according to him if you want to find out power on root of complex number let’s say it is n so n is any rational number it means it can be whole number or it can be a fraction so cos theta plus I sine theta is a complex number and we are trying to find out power of this complex number so demova says whenever you are finding the power of a complex number that time it will become cos n theta plus I sine n theta it means your power will multiply the amplitude of complex number now let’s take some example example if you have a complex number let’s say cos theta plus I sine theta raised to five it means you want to find 5th power of this complex number so according to D moyers it will become cos of five theta plus I sine v theta suppose let’s say it is – it means if we have cos theta plus I sine theta raised to minus 5 then what will happen same reason this minus Phi will multiply the amplitude of the complex number so it will become cos of minus Phi theta plus I sine minus phi theta now we all know that cos of minus theta is cos theta hence cos of minus PI theta is yes it is cos 5 theta similarly sine of minus PI theta is minus sine phi theta so it will become minus I sine 5 theta so it means if you want to find out negative power of any positive complex number then the sign may become negative and vice versa is also true it means cos phi theta minus I sine Phi that phi theta is given to you then you can write it down as cos theta plus I sine theta whole raise to minus 5 now let’s say let’s take some fraction so if it is cos theta plus I sine theta raised to 2 by 3 in this case what will happen same rule 2 by 3 it will multiply this theta so it will become cos of 2 by 3 theta plus I sine 2 by 3 theta so according to the Moira’s whatever is the power or the route that you are finding for any complex number that power or root will multiply amplitude of the complex number so using this theorem we will see some numericals and let’s solve some numericals using the result of the mobis