You can put this solution on YOUR website! Any pair of binomials (two term expressions) of the form (a+b) and (a-b) are called conjugates. And the product of conjugates is: . Note that the right side is made up of perfect square terms. Understanding this is the key to problems like yours.
In your problem you have a binomial which includes "i" in the denominator. But "i" is a square root. It is the square root of -1. And we do not want square roots of any kind in denominators. We want only rational denominators, And the easiest way to eliminate square roots in binomial denominators is to use conjugates. Just multiply the numerator and denominator by the denominator's conjugate. (Note that this procedure works for all square roots, not just for "i".)
Your denominator is 3-i. Its conjugate would be 3+i. So we will multiply the numerator and denominator by 3+i:
We now have a rational denominator, 10! If we want this complex number in a + bi form we start by splitting this into two fractions:
If you have trouble seeing this, think of it as undoing an addition. Doesn't the right side add up to the left side?
The a + bi form has some Real number, "b", times i not i divided by some number so we need to do one more thing. This is relatively easy since we can always turn a division by some number into a multiplication by the reciprocal of that number. Since the reciprocal of 10 is 1/10:
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