Simplify the following expression: $\frac{-8-\sqrt{-9}}{6}$. Give your answer in the form $a+b i$.
Answer (2)
Simplify the square root: − 9 = 3 i .
Substitute into the expression: 6 − 8 − 3 i .
Separate real and imaginary parts: 6 − 8 − 6 3 i .
Simplify: − 3 4 − 2 1 i . The final answer is − 3 4 − 2 1 i .
Explanation
Understanding the Problem We are asked to simplify the expression 6 − 8 − − 9 and express the result in the form a + bi , where a and b are real numbers. This involves dealing with the square root of a negative number, which introduces the imaginary unit i .
Simplifying the Square Root First, let's simplify the square root of -9. Recall that − 1 = i . Therefore, we can rewrite − 9 as follows: − 9 = 9 × − 1 = 9 × − 1 = 3 i
Substituting Back into the Expression Now, substitute 3 i back into the original expression: 6 − 8 − − 9 = 6 − 8 − 3 i
Separating Real and Imaginary Parts Next, we separate the real and imaginary parts by dividing both terms in the numerator by 6: 6 − 8 − 3 i = 6 − 8 − 6 3 i
Simplifying Fractions Finally, simplify the fractions: 6 − 8 = − 3 4 6 − 3 i = − 2 1 i So, the simplified expression is: − 3 4 − 2 1 i
Final Answer Thus, the expression 6 − 8 − − 9 simplified to − 3 4 − 2 1 i , which is in the form a + bi , where a = − 3 4 and b = − 2 1 .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. For example, when analyzing AC circuits, impedance (a measure of opposition to current) is often expressed as a complex number. The real part represents resistance, and the imaginary part represents reactance. By using complex numbers, engineers can easily calculate voltage and current in these circuits, ensuring efficient and safe designs.
The expression 6 − 8 − − 9 simplifies to − 3 4 − 2 1 i . This result is in the form a + bi , with a = − 3 4 and b = − 2 1 .
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