document.write( "Question 1204850: Find :
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Algebra.Com's Answer #841555 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "Find :
\n" ); document.write( "1) (- 4)^(6/4)
\n" ); document.write( "2) (root[6]((-8))^2
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\n" ); document.write( "\n" ); document.write( "Although the problem is silent about it, it is clear that in both cases the question is \r
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document.write( "            Find all values of these expressions in complex domain.\r\n" );
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\n" ); document.write( "\n" ); document.write( "So, I will solve the problem (part 1) in this formulation.\r
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\n" ); document.write( "\n" ); document.write( "Also, it is clear that the problem is not for a school student, because a normal school student does not understand
\n" ); document.write( "its meaning (together with his/her teacher) - it is for, let say, a University level students majoring in Math,
\n" ); document.write( "who study (or just studied) complex numbers analysis at the proper level.\r
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document.write( "               Question 1\r\n" );
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document.write( "(-4)^(6/4) = (-4)^(3/2) = ((-4)^3)^(1/2) =  = .\r\n" );
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document.write( "This square root of the negative number -64 has 2 (TWO) values in complex numbers.\r\n" );
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document.write( "One value is 8i.  Another value -8i.\r\n" );
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document.write( "ANSWER.  (- 4)^(6/4) has two values in complex domain.  \r\n" );
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document.write( "         One value is 8i.  Another value is -8i.\r\n" );
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\n" ); document.write( "\n" ); document.write( "So, when other tutors says that 8i is the solution, it is correct, but only partly correct.\r
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\n" ); document.write( "\n" ); document.write( "Full complete correct answer is that there are TWO values, 8i and -8i.\r
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\n" ); document.write( "\n" ); document.write( "Regarding second question, there is a FATAL ERROR in its formulation: the parentheses are UNBALANCED,
\n" ); document.write( "so the meaning of the formula is not determined and CAN NOT be determined without a fundamental repair.\r
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\n" ); document.write( "\n" ); document.write( "The lesson to learn from my post to question 1 is that in complex domain taking square root \r
\n" ); document.write( "\n" ); document.write( "is not an operation \"one value to one value\". It is an operation \"one value to TWO values\".\r
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\n" ); document.write( "\n" ); document.write( "Similar as to cubic root in complex domain is not an operation \"one value to one value\".
\n" ); document.write( "It is an operation \"one value to three values\".\r
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\n" ); document.write( "\n" ); document.write( "This understanding is one of basic conceptions, that a University student should
\n" ); document.write( "get from the University class on complex analysis (\"on complex numbers\").\r
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\n" ); document.write( "\n" ); document.write( "In reality, only small part of University students really gets this understanding.\r
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\n" ); document.write( "\n" ); document.write( "Those who really get such understanding, are either those who become professional mathematicians
\n" ); document.write( "of the \"professorship level\" with peer-reviewed articles/textbooks/monographs in the area,
\n" ); document.write( "or theoretical physicists, who use \"phases\" in their work.\r
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\n" ); document.write( "\n" ); document.write( "Of the school teachers, 99.9% of them do not understand it even at conceptual level - \r
\n" ); document.write( "\n" ); document.write( "- therefore, it is useless even to discuss this matter with them.\r
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