document.write( "Question 1197767: Which of the following is not a fourth root of -8 + 8i*square root of 3
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\n" ); document.write( "D. -square root of 3 -i
\n" ); document.write( "E. 1 - i*square root of 3 \n" ); document.write( "
Algebra.Com's Answer #831160 by ikleyn(52790)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Which of the following is not a fourth root of -8 + 8i*square root of 3
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\n" ); document.write( "D. -square root of 3 -i
\n" ); document.write( "E. 1 - i*square root of 3
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document.write( "The number \"-8+%2B+8i%2Asqrt%283%29\"  has the modulus of 16 and the argument of 120°.\r\n" );
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document.write( "Hence, the fourth roots of this number have the modulus 2 and arguments 30°, 120°, 210° and 300°.\r\n" );
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document.write( "Number  A = \"sqrt%283%29+-+i\" has the modulus 2 and the argument arg(A) = 330°.\r\n" );
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document.write( "        So, number A is NOT a fourth degree root of the given number.\r\n" );
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document.write( "Number  B = \"sqrt%283%29+%2B+i\" has the modulus 2 and the argument arg(B) = 30°.\r\n" );
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document.write( "        So, number B is one of the fourth degree roots of the given number.\r\n" );
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document.write( "Number  C = \"-1+%2B+i%2Asqrt%283%29\" has the modulus 2 and the argument arg(C) = 120°.\r\n" );
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document.write( "        So, number C is one of the fourth degree root of the given number.\r\n" );
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document.write( "Number  D = \"-sqrt%283%29+-+i\" has the modulus 2 and the argument arg(D) = 210°.\r\n" );
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document.write( "        So, number D is one of the fourth degree roots of the given number.\r\n" );
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document.write( "Number  E = \"1+-+i%2Asqrt%283%29\" has the modulus 2 and the argument arg(E) = 300°.\r\n" );
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document.write( "        So, number E is one of the fourth degree roots of the given number.\r\n" );
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document.write( "ANSWER.  There is only one number from the list which is NOT a fourth degree root of \"-8+%2B+8i%2Asqrt%283%29\": it is number (A).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, with explanations.\r
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\n" ); document.write( "\n" ); document.write( "In my solution,  I freely use properties of the roots and degrees of complex numbers  (like de Moivre theorem
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\n" ); document.write( "\n" ); document.write( "It is because this knowledge is a  PRE-Requisite for solution of this problem - so, it is assumed
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\n" ); document.write( "\n" ); document.write( "On complex numbers,  see my lessons in this site\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "    - How to take a square root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r
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\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problem on taking square root of complex number\r
\n" ); document.write( "\n" ); document.write( "    - Miscellaneous problems on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problem on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on de'Moivre formula\r
\n" ); document.write( "\n" ); document.write( "    - Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°) \r
\n" ); document.write( "\n" ); document.write( "    - A curious example of an equation in complex numbers which HAS NO a solution\r
\n" ); document.write( "\n" ); document.write( "    - Solving one non-standard equation in complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Determine locus of points using complex numbers \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Complex numbers\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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