Questions: 3. Simplify the following expressions to (a+bi) using complex number "rationalization." You may also write your answer as a single fraction so long as the denominator is a real number. (5 points each- 25 points total) (a) [ frac2-3 i ]

Solution Steps

To simplify \(\frac{2}{-3i}\), we need to eliminate the imaginary unit \(i\) from the denominator. This can be done by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(-3i\) is \(3i\).

\[ \frac{2}{-3i} \times \frac{3i}{3i} = \frac{2 \cdot 3i}{-3i \cdot 3i} \]

Multiply the numerators and denominators:

\[ \frac{6i}{-9i^2} \]

Recall that \(i^2 = -1\), so:

\[ \frac{6i}{-9(-1)} = \frac{6i}{9} \]

Divide the numerator and the denominator by 3:

\[ \frac{6i}{9} = \frac{2i}{3} \]

Final Answer