Perform the indicated operations, and write the result in the form $a+b i$. (Simplify your answers completely.)
$\frac{4-3 i}{5+4 i}$
Answer (2)
Multiply the numerator and denominator by the conjugate of the denominator: 5 + 4 i 4 − 3 i ⋅ 5 − 4 i 5 − 4 i .
Expand the numerator: ( 4 − 3 i ) ( 5 − 4 i ) = 8 − 31 i .
Expand the denominator: ( 5 + 4 i ) ( 5 − 4 i ) = 41 .
Divide to get the final answer: 41 8 − 41 31 i .
Explanation
Understanding the Problem We are asked to perform the division of two complex numbers and express the result in the form a + bi , where a and b are real numbers. The given expression is 5 + 4 i 4 − 3 i .
Multiplying by the Conjugate To divide complex numbers, we multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 5 + 4 i is 5 − 4 i . So, we multiply the given expression by 5 − 4 i 5 − 4 i .
Setting up the Multiplication 5 + 4 i 4 − 3 i × 5 − 4 i 5 − 4 i = ( 5 + 4 i ) ( 5 − 4 i ) ( 4 − 3 i ) ( 5 − 4 i )
Expanding the Numerator Now, we expand the numerator and the denominator separately.
Numerator: ( 4 − 3 i ) ( 5 − 4 i ) = 4 ( 5 ) + 4 ( − 4 i ) − 3 i ( 5 ) − 3 i ( − 4 i ) = 20 − 16 i − 15 i + 12 i 2 Since i 2 = − 1 , we have: 20 − 16 i − 15 i − 12 = 8 − 31 i
Expanding the Denominator Denominator: ( 5 + 4 i ) ( 5 − 4 i ) = 5 ( 5 ) + 5 ( − 4 i ) + 4 i ( 5 ) + 4 i ( − 4 i ) = 25 − 20 i + 20 i − 16 i 2 Since i 2 = − 1 , we have: 25 + 16 = 41
Dividing the Result Now, we divide the numerator by the denominator: 41 8 − 31 i = 41 8 − 41 31 i
Final Answer The result is in the form a + bi , where a = 41 8 and b = − 41 31 .
Examples
Complex number arithmetic is used extensively in electrical engineering, particularly in AC circuit analysis. For example, impedances of circuit elements (resistors, capacitors, and inductors) are represented as complex numbers. Calculating the total impedance of a circuit often involves dividing complex numbers, just like in this problem. This allows engineers to determine the current and voltage relationships in AC circuits, which is crucial for designing and analyzing electrical systems.
To divide the complex numbers 5 + 4 i 4 − 3 i , we multiply by the conjugate of the denominator, which results in 41 8 − 41 31 i after simplification. This process secures a result in the form a + bi . The final answer is 41 8 − 41 31 i .
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