I was wondering if someone can check my solutions and perhaps give me a faster more logical way of working through this question. Thanks. 1. The problem statement, all variables and given/known data If z = 1 + i*root3 i) Find the modulus and argument of z ii) Express z^5 in Cartesian form a + ib where a and b are real iii) Find z*zbar iv) hence, find zbar to the power of negative 5 If w = root2 cis (π/4) i) Find w/z in polar form. ii) Express z and w in cartesian form and hence find w/z in cartesian form. iii) Use answers from i) and ii) to deduce exact value for cos(π/12) 3. The attempt at a solution i) Modulus is 2, Argument is π/3 ii) Using De Moivre's Theorem, 2^5cis(5*(π/3)) = 32cis(-pie/3) Then changing to Cartesian form - 32 (cos(-π/3) + isin(-π/3)) = 32(0.5 + (root3/2)i) = 16 + 16i*root3 iii) z = 1 + i*root3, zbar = 1 - i*root3 z*zbar = 4 iv) Now this part I don't understand the "hence", as in how am I meant to use previous results to get this answer? I try - zbar = 2cis(-π/3) -> (Is it true that the "bar" of any complex number is just the same modulus and negative angle?) Then zbar^-5 = Using DM Theorem, (1/32)cis(5π/3) = (1/32)cis(-π/3) Then convert to cartesian - (1/32)(cos(-π/3) + isin(-π/3)) = (1/32)(0.5 + iroot3/2) = 1/64 + iroot3/64 If w = root2 cis (π/4) i) Find w/z in polar form. I found it in cartesian - (1+i)/(1 + root3 i) then realising, gives (1 + root3)/4 + (1-root3)/4 * i But how do I change to Polar form? I am also stuck on how to get cos(pie/12) as exact value. Thanks.