Which of the following is the correct notation for the complex number [tex]$113-\sqrt{-68}$[/tex]?
[tex]$113-2 i \sqrt{-17}$[/tex]
[tex]$113-2 \sqrt{17 i}$[/tex]
[tex]$113+2 i \sqrt{17}$[/tex]
[tex]$113-2 i \sqrt{17}$[/tex]
Answer (1)
Rewrite the complex number in the form a + bi .
Simplify the square root of the negative number: − 68 = 68 i = 2 i 17 .
Substitute the simplified term back into the original expression.
The correct notation is 113 − 2 i 17 .
Explanation
Understanding Complex Numbers We are given the complex number 113 − − 68 and asked to express it in the correct notation. A complex number is generally expressed in the form a + bi , where a and b are real numbers, and i is the imaginary unit, defined as i = − 1 .
Simplifying the Square Root First, we simplify the term − 68 . We can rewrite this as 68 ⋅ − 1 = 68 ⋅ − 1 . Since − 1 = i , we have − 68 = 68 i .
Factoring and Simplifying Now, we simplify 68 . We can factor 68 as 68 = 4 ⋅ 17 = 2 2 ⋅ 17 . Therefore, 68 = 2 2 ⋅ 17 = 2 17 .
Combining the Terms Substituting this back into our expression, we have − 68 = 2 17 i = 2 i 17 .
Final Result Finally, we substitute this back into the original complex number: 113 − − 68 = 113 − 2 i 17 .
Examples
Complex numbers are used extensively in electrical engineering to analyze AC circuits. For example, the impedance of a circuit, which is the opposition to the flow of current, can be represented as a complex number. The real part of the impedance represents the resistance, and the imaginary part represents the reactance. By using complex numbers, engineers can easily calculate the voltage and current in AC circuits.
