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sixth root of (-1+sqrt3i)^3\r
\n" ); document.write( "\n" ); document.write( "We have (a+b)^3 = (a)^3+(b)^3+3ab(a+b)
\n" ); document.write( "Here a= (-1) therefore a^3 = (-1)^3 = -1
\n" ); document.write( "b = (i sqrt3) therefore b^3 = (isqrt3)^3 = (i)^3(sqrt3)^3= (i^2Xi)X(3)X(sqrt3)= -3i(sqrt3)
\n" ); document.write( "3ab(a+b)= 3a^2b + 3ab^2
\n" ); document.write( " = 3X(-1)^2X(i sqrt3) + 3X(-1)X(i sqrt3)^2
\n" ); document.write( "=3X1X(i sqrt3)-3X(i^2)X(3)
\n" ); document.write( "=3i(sqrt3)-3X(-1)X3
\n" ); document.write( "=3i(sqrt3)+9
\n" ); document.write( "Therefore
\n" ); document.write( "(-1+sqrt3i)^3= (-1)^3 +( isqrt3)^3+3X(-1)^2X(i sqrt3) + 3X(-1)X(i sqrt3)^2
\n" ); document.write( "= -1-3i(sqrt3)+3i(sqrt3)+9
\n" ); document.write( "=-1+0+9 = 8
\n" ); document.write( "Sixth root of (-1+sqrt3i)^3 = sixth root of (8)
\n" ); document.write( "= (8)^(1/6)
\n" ); document.write( "= [(2^3)]^(1/6)
\n" ); document.write( "=(2)^(3X1/6)
\n" ); document.write( "=(2)^(1/2)
\n" ); document.write( "=sqrt(2)
\n" ); document.write( "Answer: sqrt(2)
\n" ); document.write( "Note: If you are in a class which has not yet introduced you to the concept of finding the nth roots of a complex number using trigonometric representation for a complex number and DeMoivre's Theorem, the above answer will do. Please ask for the theory of the above mentioned concept and examples if you are in a high school class. \n" ); document.write( "