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Chapter 1: Problem 75

Perform the indicated operations and write the result in standard form. $$ \frac{4}{(2+i)(3-i)} $$

Short Answer

Expert verified
The simplified expression in standard form will be the result of the operation.

Step by step solution

01

Compute Denominator鈥檚 Conjugate

The conjugate of a complex number \(a+bi\) is \(a-bi\), and the conjugate of \(a-bi\) is \(a+bi\). For the denominator, the complex numbers are \(2+i\) and \(3-i\). Therefore, the conjugates are \(2-i\) and \(3+i\) respectively.
02

Clear the Denominator

Multiply the numerator and denominator by the conjugate of the denominator. Also, remember the FOIL method (First, Outside, Inside, Last) when multiplying two binomials. Hence,\[ \frac{4}{(2+i)(3-i)}\] becomes \[ \frac{4(2-i)(3+i)} {(2+i)(2-i)(3-i)(3+i)}.\] The multiplication of a complex number with its conjugate results in the square of the magnitude of the number, which is a real number.
03

Simplify The Expression

Simplify the equation further, and compute the real and imaginary parts separately. After simplifying, the expression will be in standard form \(a+bi\) where \(a\) and \(b\) will be real numbers.

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