<PRE><B>Question 65436</B>
&lt;pre&gt;&lt;font size = 5&gt;&lt;b&gt;                ____________  
(-3i&lt;sup&gt;2&lt;/sup&gt; + 4i - 5)&lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;-9 - 4 + 5i&lt;sup&gt;3&lt;/sup&gt;

Combine the -9 and the -4 under the radical

                _________  
(-3i&lt;sup&gt;2&lt;/sup&gt; + 4i - 5)&lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;-13 + 5i&lt;sup&gt;3&lt;/sup&gt;


Since  i&lt;sup&gt;2&lt;/sup&gt; = -1 and i&lt;sup&gt;3&lt;/sup&gt; = -i, replace those
                    ___________
( -3(-1) + 4i - 5 )&lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;-13 + 5(-i)
              ________
( 3 + 4i - 5)&lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;-13 - 5i
            ________  
( -2 + 4i )&lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;-13 - 5i

Assume this equals to the complex number A + Bi, where
A and B are real numbers

          ________  
(-2 + 4i)&lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;-13 - 5i = A + Bi

Square both sides:

            ________  
(-2 + 4i)&lt;sup&gt;2&lt;/sup&gt;(&lt;font face = &quot;symbol&quot;&gt;Ö&lt;/font&gt;-13 - 5i)&lt;sup&gt;2&lt;/sup&gt; = (A + Bi)&lt;sup&gt;2&lt;/sup&gt;

(4 - 16i + 16i&lt;sup&gt;2&lt;/sup&gt;)(-13 - 5i) = A&lt;sup&gt;2&lt;/sup&gt; + 2ABi + B&lt;sup&gt;2&lt;/sup&gt;i&lt;sup&gt;2&lt;/sup&gt;

Replace the i&lt;sup&gt;2&lt;/sup&gt;&#39;s by -1

( 4 - 16i + 16(-1) )(-13 - 5i) = A&lt;sup&gt;2&lt;/sup&gt; + 2ABi + B&lt;sup&gt;2&lt;/sup&gt;(-1)

( 4 - 16i - 16 )(-13 - 5i) = A&lt;sup&gt;2&lt;/sup&gt; + 2ABi - B&lt;sup&gt;2&lt;/sup&gt;

(-12 - 16i)(-13 - 5i) = A² + 2ABi - B&lt;sup&gt;2&lt;/sup&gt;

156 + 60i + 208i + 80i&lt;sup&gt;2&lt;/sup&gt; = A&lt;sup&gt;2&lt;/sup&gt; + 2ABi - B&lt;sup&gt;2&lt;/sup&gt;

Replace the i&lt;sup&gt;2&lt;/sup&gt; by -1

156 + 60i + 208i + 80(-1) = A&lt;sup&gt;2&lt;/sup&gt; + 2ABi - B&lt;sup&gt;2&lt;/sup&gt;

156 + 60i + 208i - 80 = A&lt;sup&gt;2&lt;/sup&gt; + 2ABi - B&lt;sup&gt;2&lt;/sup&gt;

76 + 268i = A&lt;sup&gt;2&lt;/sup&gt; + 2ABi - B&lt;sup&gt;2&lt;/sup&gt;

The real numbers on the left must equal the real
numbers on the right, so

76 = A&lt;sup&gt;2&lt;/sup&gt; - B&lt;sup&gt;2&lt;/sup&gt;

A&lt;sup&gt;2&lt;/sup&gt; - B&lt;sup&gt;2&lt;/sup&gt; = 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:

A&lt;sup&gt;2&lt;/sup&gt; - B&lt;sup&gt;2&lt;/sup&gt; = 76
     AB = 134

To make things easier square the second equation

    A&lt;sup&gt;2&lt;/sup&gt;B&lt;sup&gt;2&lt;/sup&gt; = 17956

Solve it for B&lt;sup&gt;2&lt;/sup&gt;

      B&lt;sup&gt;2&lt;/sup&gt; = 17956/A&lt;sup&gt;2&lt;/sup&gt;

Substitute for B&lt;sup&gt;2&lt;/sup&gt; in the first equation

A&lt;sup&gt;2&lt;/sup&gt; - 17956/A&lt;sup&gt;2&lt;/sup&gt; = 76

A&lt;sup&gt;4&lt;/sup&gt; - 17956 = 76A&lt;sup&gt;2&lt;/sup&gt;

A&lt;sup&gt;4&lt;/sup&gt; - 76A&lt;sup&gt;2&lt;/sup&gt; - 17956 = 0

Solve that for A&lt;sup&gt;2&lt;/sup&gt; by the quadratic formula

A&lt;sup&gt;2&lt;/sup&gt; = 177.2838828, A&lt;sup&gt;2&lt;/sup&gt; = -101.2838828

But since A is real A&lt;sup&gt;2&lt;/sup&gt; is positive so we discard
the negative value

A&lt;sup&gt;2&lt;/sup&gt; = 177.2838828

Taking square roots:

A = ±13.31479939

Substitute 177.2839928 for A&lt;sup&gt;2&lt;/sup&gt; in

A&lt;sup&gt;2&lt;/sup&gt; - B&lt;sup&gt;2&lt;/sup&gt; = 76

177.2838828 - B&lt;sup&gt;2&lt;/sup&gt; = 76

177.2838828 - 76 = B&lt;sup&gt;2&lt;/sup&gt;

101.2838828 = B&lt;sup&gt;2&lt;/sup&gt;

B&lt;sup&gt;2&lt;/sup&gt; = 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&lt;/pre&gt;


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