Find the multiplicative inverse of 2−3i.
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Step-by-Step Solution
Step 1: Define Multiplicative Inverse
The multiplicative inverse of a complex number z is denoted as z−1 or z1. For a complex number z=a+bi, its multiplicative inverse is a+bi1.
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)=a2−b2.
Step 5: Simplify the Denominator
Recall that i2=−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.