Simplify. Write your response in $a+b i$ form.
$\frac{-3-8 i}{-7+10 i}=$
Answer (2)
Multiply the numerator and denominator by the conjugate of the denominator.
Expand the numerator: ( − 3 − 8 i ) ( − 7 − 10 i ) = − 59 + 86 i .
Expand the denominator: ( − 7 + 10 i ) ( − 7 − 10 i ) = 149 .
Simplify the expression: − 7 + 10 i − 3 − 8 i = − 149 59 + 149 86 i . The final answer is − 149 59 + 149 86 i .
Explanation
Understanding the Problem We are asked to simplify the complex number expression − 7 + 10 i − 3 − 8 i and write the result in the form a + bi , where a and b are real numbers.
Multiplying by the Conjugate To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of − 7 + 10 i is − 7 − 10 i . Thus, we have − 7 + 10 i − 3 − 8 i = − 7 + 10 i − 3 − 8 i ⋅ − 7 − 10 i − 7 − 10 i .
Expanding the Numerator Now, we multiply out the numerator: ( − 3 − 8 i ) ( − 7 − 10 i ) = ( − 3 ) ( − 7 ) + ( − 3 ) ( − 10 i ) + ( − 8 i ) ( − 7 ) + ( − 8 i ) ( − 10 i ) = 21 + 30 i + 56 i + 80 i 2 . Since i 2 = − 1 , we have 21 + 30 i + 56 i − 80 = − 59 + 86 i .
Expanding the Denominator Next, we multiply out the denominator: ( − 7 + 10 i ) ( − 7 − 10 i ) = ( − 7 ) ( − 7 ) + ( − 7 ) ( − 10 i ) + ( 10 i ) ( − 7 ) + ( 10 i ) ( − 10 i ) = 49 + 70 i − 70 i − 100 i 2 . Since i 2 = − 1 , we have 49 + 100 = 149.
Simplifying the Expression Therefore, we have − 7 + 10 i − 3 − 8 i = 149 − 59 + 86 i = − 149 59 + 149 86 i .
Final Answer The simplified form is a + bi where a = − 149 59 and b = 149 86 .
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. The impedance, which is the opposition to the flow of current in an AC circuit, is a complex quantity. Simplifying expressions involving complex numbers, like the one in this problem, is essential for analyzing and designing AC circuits. For example, calculating the total impedance of a circuit often involves dividing complex numbers, and expressing the result in the standard a + bi form allows engineers to easily determine the real (resistance) and imaginary (reactance) components of the impedance.
To simplify − 7 + 10 i − 3 − 8 i , we multiply by the conjugate of the denominator, resulting in the form − 149 59 + 149 86 i . This expression is in the standard form of complex numbers, a + bi .
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