____________
(-3i2 + 4i - 5)Ö-9 - 4 + 5i3
Combine the -9 and the -4 under the radical
_________
(-3i2 + 4i - 5)Ö-13 + 5i3
Since i2 = -1 and i3 = -i, replace those
___________
( -3(-1) + 4i - 5 )Ö-13 + 5(-i)
________
( 3 + 4i - 5)Ö-13 - 5i
________
( -2 + 4i )Ö-13 - 5i
Assume this equals to the complex number A + Bi, where
A and B are real numbers
________
(-2 + 4i)Ö-13 - 5i = A + Bi
Square both sides:
________
(-2 + 4i)2(Ö-13 - 5i)2 = (A + Bi)2
(4 - 16i + 16i2)(-13 - 5i) = A2 + 2ABi + B2i2
Replace the i2's by -1
( 4 - 16i + 16(-1) )(-13 - 5i) = A2 + 2ABi + B2(-1)
( 4 - 16i - 16 )(-13 - 5i) = A2 + 2ABi - B2
(-12 - 16i)(-13 - 5i) = A² + 2ABi - B2
156 + 60i + 208i + 80i2 = A2 + 2ABi - B2
Replace the i2 by -1
156 + 60i + 208i + 80(-1) = A2 + 2ABi - B2
156 + 60i + 208i - 80 = A2 + 2ABi - B2
76 + 268i = A2 + 2ABi - B2
The real numbers on the left must equal the real
numbers on the right, so
76 = A2 - B2
A2 - B2 = 76
The imaginary numbers on the left must equal the
imaginary numbers on the right, so
268i = 2ABi
Dividing thru by 2i
134 = AB
AB = 134
So we have this system of equations:
A2 - B2 = 76
AB = 134
To make things easier square the second equation
A2B2 = 17956
Solve it for B2
B2 = 17956/A2
Substitute for B2 in the first equation
A2 - 17956/A2 = 76
A4 - 17956 = 76A2
A4 - 76A2 - 17956 = 0
Solve that for A2 by the quadratic formula
A2 = 177.2838828, A2 = -101.2838828
But since A is real A2 is positive so we discard
the negative value
A2 = 177.2838828
Taking square roots:
A = ±13.31479939
Substitute 177.2839928 for A2 in
A2 - B2 = 76
177.2838828 - B2 = 76
177.2838828 - 76 = B2
101.2838828 = B2
B2 = 101.2838828
Taking square roots:
B = ± 10.06398941
So we get four solutions:
A + Bi = 13.31479939 + 10.06398941i
A + Bi = 13.31479939 - 10.06398941i
A + Bi = -13.31479939 + 10.06398941i
A + Bi = -13.31479939 - 10.06398941i
However we can eliminate two of these because
of the equation
AB = 134
since 134 is a positive number, which means
that A and B must have the same sign. This
only leaves the two solutions:
A + Bi = 13.31479939 + 10.06398941i
A + Bi = -13.31479939 - 10.06398941i
Both those are correct answers because
there are two complex imaginary square
roots of a complex imaginary number.
The original problem contains
a square root of a complex number, thus
we expect two answers.
Edwin
