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Express five plus five π times eight minus four π over one minus π in the form π plus ππ.
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In this question, weβre dividing the product of two complex numbers by another complex number.
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And so we recall how we divide by complex numbers.
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We write the quotient in fraction form as in this question.
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And then we multiply both the numerator and denominator of our fraction by the conjugate of the denominator.
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And when we talk about the conjugate, weβre essentially changing the sign of the imaginary part of our complex number.
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So if we have the general form π§ as being π plus ππ, the conjugate which we denote as π§ bar is π minus ππ.
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So if we take the quotient here, the conjugate of one minus π is one plus π.
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So weβre going to multiply both the numerator and denominator of our fraction by one plus π.
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Letβs begin by calculating the denominator.
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Thatβs one minus π times one plus π.
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We multiply the first number in each expression.
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One times one is equal to one.
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Then we multiply the outer terms, and we get one times π, which is simply π.
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Next, we multiply the inner terms.
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Negative π times one is negative π.
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And finally, we multiply the last term in each expression, and we get negative π squared.
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Now, π minus π is zero, and we actually know that π squared is equal to negative one.
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So this simplifies to one minus negative one.
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Well, thatβs one plus one, which is just two.
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But what do we do with our numerator?
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Well, weβre going to do this in stages.
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Weβll just begin by distributing the parentheses containing five plus five π and eight minus four π.
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In the same way, we multiply the first terms to get 40.
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We multiply the outer terms.
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Five times negative four π is negative 20π.
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We have five π times eight, which is 40π, and then five π times negative four π, which is negative 20π squared.
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Then negative 20π plus 40π is positive 20π.
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But we can also rewrite negative 20π squared as negative 20 times negative one.
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Well, a negative times a negative is a positive.
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So we get 40 plus 20 as being 60.
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And then we find the product of five plus five π and eight minus four π is 60 plus 20π.
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Letβs now multiply all of this by one plus π.
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Distributing as before, and we get 60 plus 60π plus 20π plus 20π squared.
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60π plus 20π is 80π, but also 20 times π squared is 20 times negative one.
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So itβs negative 20.
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And so we simplify this expression fully, and we get 40 plus 80π.
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And so when we divide five plus five π times eight minus four π by one minus π, we get 40 plus 80π over two, which we can simplify further by dividing each term on the numerator by two.
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40 divided by two is 20, and then 80π divided by two is 40π.
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And so by multiplying the numerator and the denominator of our fraction by the conjugate of the denominator, weβve expressed it in the form π plus ππ.
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Itβs 20 plus 40π.