Simplify. Write your response in $a+b i$ form.
$\frac{11-6 i}{-11 i}=$
Answer (2)
Multiply the numerator and denominator by i .
Distribute and simplify the expression using i 2 = − 1 .
Separate the real and imaginary parts.
The simplified form is 11 6 + i .
Explanation
Understanding the Problem We are asked to simplify the complex number − 11 i 11 − 6 i and express it in the standard form a + bi , where a and b are real numbers.
Multiplying by i To simplify the given expression, we can multiply the numerator and the denominator by i to eliminate the imaginary unit from the denominator: − 11 i 11 − 6 i = ( − 11 i ) × i ( 11 − 6 i ) × i
Simplifying the Expression Now, we distribute i in the numerator and simplify the denominator: − 11 i 2 11 i − 6 i 2 Since i 2 = − 1 , we substitute this into the expression: − 11 ( − 1 ) 11 i − 6 ( − 1 ) = 11 11 i + 6
Final Answer Finally, we write the complex number in the standard form a + bi :
11 6 + 11 i = 11 6 + 11 11 i = 11 6 + i So, the simplified form of the given complex number is 11 6 + i .
Examples
Complex numbers are used in electrical engineering to analyze AC circuits. For example, the impedance of a circuit can be represented as a complex number, where the real part represents resistance and the imaginary part represents reactance. Simplifying complex numbers helps engineers to understand and design these circuits effectively.
The expression − 11 i 11 − 6 i can be simplified to 11 6 + i by multiplying the numerator and denominator by i , simplifying, and separating real and imaginary parts.
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