Simplify. Write your response in $a+b i$ form.
$\frac{7-2 i}{-9+5 i}=$
$\square$
Answer (1)
Multiply the numerator and denominator by the conjugate of the denominator.
Expand the numerator: ( 7 − 2 i ) ( − 9 − 5 i ) = − 73 − 17 i .
Expand the denominator: ( − 9 + 5 i ) ( − 9 − 5 i ) = 106 .
Simplify the fraction: − 106 73 − 106 17 i .
Explanation
Understanding the Problem We are asked to simplify the complex fraction − 9 + 5 i 7 − 2 i and express the result in the form a + bi , where a and b are real numbers.
Finding the Conjugate To simplify the complex fraction, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of − 9 + 5 i is − 9 − 5 i .
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: − 9 + 5 i 7 − 2 i × − 9 − 5 i − 9 − 5 i = ( − 9 + 5 i ) ( − 9 − 5 i ) ( 7 − 2 i ) ( − 9 − 5 i )
Expanding the Numerator Expand the numerator: ( 7 − 2 i ) ( − 9 − 5 i ) = 7 ( − 9 ) + 7 ( − 5 i ) − 2 i ( − 9 ) − 2 i ( − 5 i ) = − 63 − 35 i + 18 i + 10 i 2 = − 63 − 17 i − 10 = − 73 − 17 i
Expanding the Denominator Expand the denominator: ( − 9 + 5 i ) ( − 9 − 5 i ) = ( − 9 ) ( − 9 ) + ( − 9 ) ( − 5 i ) + ( 5 i ) ( − 9 ) + ( 5 i ) ( − 5 i ) = 81 + 45 i − 45 i − 25 i 2 = 81 + 25 = 106
Simplifying the Fraction Now, we have: 106 − 73 − 17 i = 106 − 73 − 106 17 i
Final Answer Thus, the simplified form of the complex fraction is 106 − 73 − 106 17 i . Approximating the real and imaginary parts to three decimal places, we have approximately − 0.689 − 0.160 i .
Conclusion Therefore, the simplified form of the given complex number is − 106 73 − 106 17 i .
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. Impedance, which is the opposition to the flow of current in an AC circuit, is a complex quantity. By simplifying complex fractions, electrical engineers can analyze and design AC circuits more effectively, ensuring proper circuit behavior and preventing damage to components. For example, calculating the equivalent impedance of a parallel circuit involves simplifying complex fractions.
