Simplify. Write your response in $a+b i$ form. [tex] \frac{-5+2 i}{11+10 i}=\square [/tex]
Answer (1)
Multiply the numerator and denominator by the conjugate of the denominator.
Expand the numerator and denominator.
Simplify the expression using i 2 = − 1 .
Divide the real and imaginary parts by the denominator: − 221 35 + 221 72 i .
Explanation
Understanding the Problem We are asked to simplify the complex fraction 11 + 10 i − 5 + 2 i and express it in the standard form a + bi , where a and b are real numbers. To do this, we need to eliminate the imaginary part from the denominator.
Using the Conjugate To eliminate the imaginary part from the denominator, we multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 11 + 10 i is 11 − 10 i .
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: 11 + 10 i − 5 + 2 i ⋅ 11 − 10 i 11 − 10 i
Expanding the Numerator Expand the numerator: ( − 5 + 2 i ) ( 11 − 10 i ) = − 5 ( 11 ) − 5 ( − 10 i ) + 2 i ( 11 ) + 2 i ( − 10 i ) = − 55 + 50 i + 22 i − 20 i 2 Since i 2 = − 1 , we have: − 55 + 50 i + 22 i + 20 = − 35 + 72 i
Expanding the Denominator Expand the denominator: ( 11 + 10 i ) ( 11 − 10 i ) = 11 ( 11 ) + 11 ( − 10 i ) + 10 i ( 11 ) + 10 i ( − 10 i ) = 121 − 110 i + 110 i − 100 i 2 Since i 2 = − 1 , we have: 121 + 100 = 221
Dividing by the Denominator Now, divide the real and imaginary parts of the numerator by the denominator: 221 − 35 + 72 i = 221 − 35 + 221 72 i
Final Answer The simplified form of the complex fraction is 221 − 35 + 221 72 i .
Examples
Complex numbers are used in electrical engineering to represent alternating currents and voltages. Simplifying complex fractions helps in analyzing circuits and understanding the relationships between different electrical components. For example, when calculating impedance in AC circuits, you often encounter complex fractions that need to be simplified to find the overall impedance of the circuit. This allows engineers to design and optimize electrical systems effectively.
