document.write( "Question 90564: Hi, how do i perform the following operation and express it in standard form?
\n" ); document.write( "2 - 6i / 4 - 9i \n" ); document.write( "

Algebra.Com's Answer #65777 by bucky(2189) $\"\"$ $\"About$

You can put this solution on YOUR website!
The discussion shows one of a couple of ways that you can do this problem.
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\n" ); document.write( "This way involves converting the denominator to a real number and then dividing that real
\n" ); document.write( "number into each of the terms in the resulting complex numerator.
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\n" ); document.write( "Suppose we find the complex conjugate of the denominator. Since the denominator is 4 - 9i, its
\n" ); document.write( "complex conjugate is the same but with the opposite sign between it real and imaginary
\n" ); document.write( "parts. In other words, the complex conjugate of the denominator is 4 + 9i.
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\n" ); document.write( "Now suppose we multiply the original problem by:
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\n" ); document.write( " $\"%284+%2B+9i%29%2F%284+%2B+9i%29\"$
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\n" ); document.write( "Notice that this fraction, composed of the complex conjugate over the complex conjugate
\n" ); document.write( "is equivalent to 1, so in effect we are multiplying the original problem by 1. This multiplication
\n" ); document.write( "is written as:
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\n" ); document.write( " $\"%28%282-6i%29%2F%284-9i%29%29%2A%28%284%2B9i%29%2F%284%2B9i%29%29\"$
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\n" ); document.write( "Notice that the multiplication of the denominators involves two terms of the form
\n" ); document.write( " $\"%28a+-+b%29%2A%28a+%2B+b%29\"$ . If you multiply out this form the answer you get is $\"a%5E2+-+b%5E2\"$
\n" ); document.write( "a form that you might recall from basic algebra. In this problem a = 4 and b = 9i, so the
\n" ); document.write( "multiplication of the two denominators results in $\"4%5E2+-+%289i%29%5E2\"$ . Squaring the 4
\n" ); document.write( "results in 16 and squaring the 9i results in $\"9%5E2i%5E2%29\"$ . But recall that the definition
\n" ); document.write( "of $\"i%5E2\"$ is that it equals -1. So when you square 9i you get 81*(-1) or -81. So the product
\n" ); document.write( "of the denominator of the original problem times its conjugate ... in other words $\"a%5E2+-+b%5E2\"$
\n" ); document.write( "becomes $\"16+-+%28-81%29\"$ which is $\"16+%2B+81+=+97\"$ . Let's not lose sight of what is
\n" ); document.write( "going on here. So far what we have done has resulted in:
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\n" ); document.write( "Observe that the resulting denominator is a real number ... it is 97. Now we need to
\n" ); document.write( "multiply $\"%282+-+6i%29%2A%284+%2B+9i%29\"$ . First multiply the 2 times (4 + 9i) to get 8 + 18i. Then
\n" ); document.write( "multiply the -6i times (4 + 9i) to get $\"-24i+-54i%5E2\"$ . Then combine the two products
\n" ); document.write( "to get:
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\n" ); document.write( " $\"8+%2B+18i+-24i+-+54i%5E2\"$
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\n" ); document.write( "The +18i and the - 24i combine to give -6i, simplifying the result to:
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\n" ); document.write( " $\"8+-+6i+-+54i%5E2\"$
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\n" ); document.write( "Again, recall that the definition of $\"i%5E2\"$ is that $\"i%5E2+=+-1\"$ . Substituting
\n" ); document.write( "-1 for $\"i%5E2\"$ results in:
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\n" ); document.write( " $\"8+-+6i+-54%28-1%29+=+8+-+6i+-%28-54%29+=+8+-+6i+%2B+54+=+62+-+6i\"$
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\n" ); document.write( "So the problem progression is now:
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\n" ); document.write( "As a final step, all that needs to be done now is to divide 97 into each of the terms of
\n" ); document.write( "the numerator and the answer becomes:
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\n" ); document.write( " $\"%2862+-+6i%29%2F97+=+62%2F97+-+6i%2F97+=+0.639175257+-+0.06185567i\"$
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\n" ); document.write( "And there's the answer.
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\n" ); document.write( "Hope this helps you to understand the problem. It's important to know that if you have a
\n" ); document.write( "complex denominator, you can convert it to a real number by multiplying it by its
\n" ); document.write( "conjugate ... but when you do that multiplication, you must also multiply the numerator
\n" ); document.write( "by that same conjugate. The rest of the problem involves just careful multiplication
\n" ); document.write( "and the recognition that $\"i%5E2+=+-1\"$ .
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