query: How can $~(z+1)^6 = z^6~$ , where $~z~$ is a complex number?
positive_passages.docid: 16400
positive_passages.text: Find all solutions to the equation $(z+1)^6=z^6$. Express the solutions in simplified rectangular form. Hint: Clearly $z=0$ is not a solution.
negative_passages.docid: 124438
negative_passages.text: Find $|(\mathbb{Z} \oplus \mathbb{Z}) / \langle(3,6)\rangle|$ and determine if its cyclic
negative_passages.docid: 66210
negative_passages.text: Prove that $z^n + 1/{z^n}$ is a rational number
negative_passages.docid: 113993
negative_passages.text: Complex power of a complex number
negative_passages.docid: 143123
negative_passages.text: finding an argument of a complex number
negative_passages.docid: 120092
negative_passages.text: Express the complex numbers given in Problem in the form $a+ib$
negative_passages.docid: 81571
negative_passages.text: Complex Exponent of Complex Numbers
negative_passages.docid: 52800
negative_passages.text: Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$
negative_passages.docid: 48337
negative_passages.text: Complex numbers equation: $z^4 = -16$
negative_passages.docid: 65416
negative_passages.text: Is there a different way to define multiplication on Complex numbers?
negative_passages.docid: 30781
negative_passages.text: Why the representation in the form of $Z'X(X'X)^{-1}X'Z$ can not be simplified into $Z'Z$
negative_passages.docid: 60614
negative_passages.text: Why is it that Complex Numbers are algebraically closed?
negative_passages.docid: 17355
negative_passages.text: How do I get the square root of a complex number?
negative_passages.docid: 24822
negative_passages.text: How to solve the equation $(z+1)^7 = z^7$ for all $z$?
negative_passages.docid: 34709
negative_passages.text: Complex numbers - roots of unity
negative_passages.docid: 122181
negative_passages.text: Application of Complex Numbers
negative_passages.docid: 34252
negative_passages.text: $ z^n = a_n + b_ni $ Show that $ b_{n+2} - 2b_{n+1} + 5b_n = 0 $ (complex numbers)