```Question 120429
Given:
.
{{{1/(4-7i)}}}
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You can convert the denominator of this fraction to a real number by multiplying the denominator
by its conjugate ... the same term as the denominator only with the opposite sign between the
terms. In this case, the conjugate is {{{4+7i}}}. If you multiply the denominator by its
conjugate, you must also multiply the numerator by the same conjugate.
.
This multiplication leads to:
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{{{1/(4-7i)*(4+7i)/(4+7i)}}}
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When you multiply the denominators:
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{{{(4-7i)*(4+7i)}}}
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You can do so by multiplying the 4 in the first set of parentheses by both terms in the second
set of parentheses and then multiplying the -7i from the first set of parentheses by
both terms in the second set of parentheses:
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{{{(4-7i)*(4+7i) = 4*4 + 4*7i -7i*4 -7i*7i= 16 + 28i -28i -49i^2}}}
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Notice that the +28i and the -28i are equal but of opposite sign. Therefore, they cancel
each other out and you are left with:
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{{{16 - 49i^2}}}
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But, by definition, {{{i^2 = -1}}}. Substitute -1 for {{{i^2}}} and the expression becomes:
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{{{16-49(-1) = 16+49 = 65}}}
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So the denominator, when multiplied by its conjugate, becomes 65.
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The numerator, when multiplied by the conjugate, is:
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{{{1*(4+7i) = 4+7i}}}
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This numerator is then over the denominator 65 to give:
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{{{(4+7i)/65 = 4/65 + (7/65)i}}}
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So the answer to this problem, in the form a + bi, is:
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{{{4/65 +(7/65)i}}}
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Hope this helps you to understand the problem.
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