When rationalizing, what must you multiply by to eliminate the imaginary part in the denominator?

When rationalizing a fraction that has a complex number in the denominator, the goal is to eliminate the imaginary part, allowing for a simpler form. To achieve this, you multiply both the numerator and denominator by the complex conjugate of the denominator.

The complex conjugate of a complex number ( a + bi ) is ( a - bi ). By multiplying by the conjugate, the imaginary unit ( i ) in the denominator is eliminated because:

[

(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2(-1) = a^2 + b^2

]

This leaves you with a real number in the denominator. Thus, if the denominator is ( a + bi ), multiplying by ( a - bi ) effectively rationalizes it and removes the imaginary component. This is why selecting ( (a - bi) ) achieves the desired effect of eliminating the imaginary part from the denominator.