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6-10 Simplifying Expressions Containing Complex Numbers As we saw in the last example from 6-9, when we multiply two imaginary numbers together, we get a real number. This being the case we can use the concept of conjugates that we have had so much experience with in the past few sections. a+bi and a-bi are conjugates. Let’s see it in action. Simplify 6 3i 6 3i . Use FOIL. 6 3i 6 3i 36 18i 18i 9i 2 36 9(1) 45 What if we interject some variables? Simplify x 2 yi x 2 yi . x 2 yi x 2 yi x 2 2 xyi 2 xyi 4 y 2 i 2 x 2 4 y 2 (1) x2 4y2 Since i is technically a radical, and since we must eliminate radicals from the denominator to fully simplify a fraction, we must also eliminate imaginary numbers. Simplify 2 8i 3i 2 8i 2 8i i 3i 3i i 2i 8i 2 3i 2 2i 8 8 2i or 3 3 If we find we have a complex number in the denominator, we can use its conjugate to clear the radical. Simplify 3 9i . 4 2i 3 9i 3 9i 4 2i 4 2i 4 2i 4 2i 12 6i 36i 18i 2 16 4i 2 12 42i 18 20 6 42i 3 21i 20 10 Last, but not least, our old friend the multiplicative inverse, rears its head. Find the multiplicative inverse of 7 2i . Simple, right? It is 1 . All done, right? 7 2i Wrong!! You can’t have radicals in the denominator. Simplify. 1 1 7 2i 7 2i 7 2i 7 2i 7 2i 49 4i 2 7 2i 53