document.write( "Question 350605: 3 roots 2i-2 \n" ); document.write( "
Algebra.Com's Answer #250617 by CharlesG2(834)\"\" \"About 
You can put this solution on YOUR website!
3 roots 2i-2\r
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\n" ); document.write( "\n" ); document.write( "solving to find the 3rd roots of 2i - 2, if this is not what you need,
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\n" ); document.write( "\n" ); document.write( "convert 2i - 2 to polar form
\n" ); document.write( "2i - 2 = -2 + 2i = x + yi = z
\n" ); document.write( "z = r * (cos(t) + i * sin(t))
\n" ); document.write( "r * (cos(t) + i * sin(t)) is polar form
\n" ); document.write( "t is the angle, theta
\n" ); document.write( "r = |z| = sqrt(x^2 + y^2) = sqrt((-2)^2 + 2^2)
\n" ); document.write( "r = sqrt((-2)^2 + 2^2), this is the magnitude of the complex number, or the distance it is from origin if plotted on a complex plane
\n" ); document.write( "r = sqrt(4 + 4) = 8^(1/2) = sqrt(8) = sqrt(4 * 2) = 2sqrt(2)
\n" ); document.write( "-2 < 0 and 2 > 0, so this is in quadrant 2 when plotted on a complex plane
\n" ); document.write( "(quadrants go counterclockwise)
\n" ); document.write( "tan t = |y/x| = |2/(-2)| = |-1| = 1
\n" ); document.write( "t = arctangent(1)
\n" ); document.write( "t = 45, since quadrant 2 --> t = 180 - 45 = 135 degrees
\n" ); document.write( "z = 2i - 2 = -2 + 2i = 2sqrt(2)(cos(135) + isin(135)) in polar form\r
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\n" ); document.write( "\n" ); document.write( "we want third roots of 2sqrt(2)(cos(135) + isin(135)),
\n" ); document.write( "360 degrees / 3 = 120 degrees, these roots will be 120 degrees apart
\n" ); document.write( "using de Moivre's formula
\n" ); document.write( "[r * (cosx + isinx)]^n = r^n * (cos(nx) + isin(nx))
\n" ); document.write( "r = 8^(1/2) = sqrt(8) = sqrt(4 * 2) = 2sqrt(2)
\n" ); document.write( "r^(1/3) = (8^(1/2)^(1/3) = 8^(1/2 * 1/3) = 8^(1/6) = (8^(1/3))^(1/2) = 2^(1/2)
\n" ); document.write( "r^(1/3) = 2^(1/2) = sqrt(2)
\n" ); document.write( "z^(1/3) = sqrt(2)[cos(1/3 * (135 + 360k)) + isin(1/3 * (135 + 360k))]
\n" ); document.write( "k from 0 to 2, since n = 3\r
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\n" ); document.write( "\n" ); document.write( "k = 0
\n" ); document.write( "sqrt(2)[cos(1/3 * 135) + isin(1/3 * 135)]
\n" ); document.write( "sqrt(2)[cos(45) + isin(45)]
\n" ); document.write( "sqrt(2)[sqrt(2)/2 + i(sqrt(2)/2)]
\n" ); document.write( "2/2 + i(2/2)
\n" ); document.write( "1 + i\r
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\n" ); document.write( "\n" ); document.write( "k = 1
\n" ); document.write( "sqrt(2)[cos(1/3 * 495) + isin(1/3 * 495)]
\n" ); document.write( "sqrt(2)[cos(165) + isin(165)]
\n" ); document.write( "sqrt(2)cos(165) + isqrt(2)sin(165)\r
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\n" ); document.write( "\n" ); document.write( "k = 2
\n" ); document.write( "sqrt(2)[cos(1/3 * 855) + isin(1/3 * 855)]
\n" ); document.write( "sqrt(2)[cos(285) + isin(285)]
\n" ); document.write( "sqrt(2)cos(285) + isqrt(2)sin(285)\r
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\n" ); document.write( "\n" ); document.write( "check:
\n" ); document.write( "(1 + i)(1 + i)(1 + i)
\n" ); document.write( "(1 + 2i - 1)(1 + i) (since i^2 = -1)
\n" ); document.write( "2i(1 + i)
\n" ); document.write( "2i + 2i^2
\n" ); document.write( "2i - 2 = -2 + 2i, yes
\n" ); document.write( "[sqrt(2)cos(165) + isqrt(2)sin(165)]^3
\n" ); document.write( "inputted (sqrt(2)cos(165) + i*sqrt(2)sin(165))^3
\n" ); document.write( "into http://www.wolframalpha.com
\n" ); document.write( "you can multiply it out if you want
\n" ); document.write( "did come out to -2 + 2i, yes
\n" ); document.write( "[sqrt(2)cos(285) + isqrt(2)sin(165)]^3
\n" ); document.write( "inputted (sqrt(2)cos(285) + i*sqrt(2)sin(285))^3
\n" ); document.write( "into http://www.wolframalpha.com
\n" ); document.write( "you can multiply it out if you want
\n" ); document.write( "did come out to -2 + 2i, yes\r
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