Find the quotient (7+4i)/(2+2i)

Question: find quotient (7+4i)/(2+2i)
Find the quotient (7+4i)/(2+2i) in simplified form.

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Answered By:

Michael M.
Caltech grad teaches Algebra 1 the way your brain wants to learn

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Written Explanation:

Find the quotient (7+4i)/(2+2i) in simplified form.As your tutor, I'll teach you both some facts that you need to have memorized, and some guiding rules of thumb or guiding intuitions that you can use to understand this problem and to solve other problems, even entirely new ones.The quotient should have the form A + Bi. That is, it should be a complex number in simplified form. This fraction may look puzzling. The guiding intuition I want you to have is that you can manipulate fractions. You don't have to feel that you are stuck with this fraction in its current form.Now, for a specific fact about manipulating a fraction: you can multiply the top and bottom by the same thing. For an example you probably first saw a long time ago, say we have the fraction 1/2. We can multiply the top and bottom by some number, let's say 3.1/2 = (1 * 3)(2 * 3) = 3/6Another fact: you can also manipulate complex numbers.A number such as 2+2i has something called a conjugate which in this case is 2-2i. I just flipped the sign on the imaginary part of the number.You should have learned about multiplying complex numbers by this point in class. If you haven't, I'll drill it with you.So, if I multiply a complex number by its conjugate, I get a real number. Say(2+2i)(2-2i) = (2)(2) + (2)(2i) + (2)(-2i) + (2i)(-2i) = 4 + 4i - 4i - 2i2 = 4 + 4 = 8Now we put these two facts together: I can multiply the top and bottom of the fraction by the conjugate of the denominator:(7+4i)(2+2i) = ((7+4i)(2-2i))/((2+2i)(2-2i)) = (22-6i)/8 = 11/4 - 3i/4Now we have a complex number with real part 11/4 and imaginary part -3i/4. This is the answer.

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