Find the multiplicative inverse of $2-3 i$.

Step 1: Define Multiplicative Inverse

The multiplicative inverse of a complex number $z$ is denoted as $z^{-1}$ or $\frac{1}{z}$. For a complex number $z = a + bi$, its multiplicative inverse is $\frac{1}{a + bi}$.

Step 2: Substitute the Complex Number

We are given the complex number $z = 2-3i$. We substitute this value into the formula for the multiplicative inverse.

Step 3: Multiply by the Conjugate

To simplify a complex fraction, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $2-3i$ is $2+3i$. This eliminates the imaginary part from the denominator.

Step 4: Perform Multiplication

Multiply the numerators and the denominators. In the denominator, we use the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$.

Step 5: Simplify the Denominator

Recall that $i^2 = -1$. Substitute this value into the denominator and simplify the expression.

Step 6: Express in Standard Form

Finally, express the result in the standard form of a complex number, $a+bi$, by separating the real and imaginary parts.

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