Question

assignment 6: problem 10
(1 point)
evaluate the expression
\\( \frac { ( 4 + 2 i ) ( - 2 i ) } { - 2 + 1 i } \\)
and write the result in the form \\( a + b i \\).
then \\( a = \\) and \\( b = \\)
note: you can earn partial credit on this problem.
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Explanation:

Step1: Simplify the numerator

Multiply \((4 + 2i)\) and \((-2i)\) using the distributive property \((a + b)c=ac+bc\).

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Since \(i^{2}=-1\), then \(-8i - 4i^{2}=-8i+4 = 4-8i\).

Step2: Rationalize the denominator

Multiply the numerator and denominator of \(\frac{4 - 8i}{-2 + i}\) by the conjugate of the denominator \(-2 - i\).

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Expand the numerator \((4-8i)(-2 - i)=4\times(-2)+4\times(-i)-8i\times(-2)-8i\times(-i)=-8-4i + 16i+8i^{2}\).
Since \(i^{2}=-1\), we have \(-8-4i + 16i-8=-16 + 12i\).
Expand the denominator \((-2 + i)(-2 - i)=(-2)^{2}-i^{2}=4+1 = 5\) (using the formula \((a + b)(a - b)=a^{2}-b^{2}\)).

Step3: Write in the form \(a+bi\)

\(\frac{-16 + 12i}{5}=\frac{-16}{5}+\frac{12}{5}i\)

Answer:

\(a =-\frac{16}{5}\), \(b=\frac{12}{5}\)