When rationalizing a denominator that is a complex number, what is multiplied by?

To rationalize a denominator that is a complex number, it is essential to eliminate the imaginary unit ( i ) from the denominator. The correct approach involves multiplying the entire expression by the conjugate of the complex number in the denominator.

In the case of a complex number in the form ( a + bi ) (where ( a ) and ( b ) are real numbers and ( i ) is the imaginary unit), multiplying by the conjugate, which is ( a - bi ), will help to simplify the expression. This is because the multiplication of a complex number by its conjugate results in a real number. Specifically, ( (a + bi)(a - bi) = a^2 + b^2 ), a non-negative real number, effectively rationalizing the denominator.

Thus, the correct choice of multiplying by the conjugate, which results in a simplified, rational denominator, confirms that option—which suggests using the form that is beneficial for achieving this result. This method is fundamental in complex number arithmetic and is crucial for simplifying expressions for further mathematical operations.