Q:

How Are Imaginary Denominators Rationalized?

A:

To rationalize an imaginary denominator, multiply both the numerator and denominator by the complex conjugate of the denominator. Multiply out the terms in the numerator and denominator, then simplify the result as much as possible. The result has imaginary numbers in the numerator, but the denominator is a real number.

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The complex conjugate of an imaginary number a + b*i is a - b*i for any real numbers a and b. Multiplying an imaginary number by its complex conjugate produces a real number because, when all terms are multiplied out, there are two real terms (a² and b²) and two imaginary terms that cancel out (+a*b*i and -a*b*i). The term b² is positive because i² is equal to -1.

After simplifying the numerator and denominator, check to see if both the real and imaginary terms in the numerator are divisible by the denominator. In such cases, the quantity can be simplified further.

Both the numerator and denominator of the original quantity must be multiplied by the complex conjugate to avoid changing the value of the quantity. By multiplying both the numerator and denominator by the complex conjugate, the quantity has essentially been multiplied by 1, and the value remains unchanged.