Write $9-\sqrt{-64}$ as a complex number.
A) $9-9 i$
B) $9-8 i$
C) $9-\sqrt{-8}$
D) $9+8 i$
Answer (1)
Simplify the square root of the negative number: − 64 = 8 i .
Substitute the simplified value back into the original expression: 9 − − 64 = 9 − 8 i .
The complex number is now in the standard form a + bi .
The final answer is 9 − 8 i .
Explanation
Understanding Complex Numbers We are asked to write 9 − − 64 as a complex number. A complex number is of 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 square root of the negative number: − 64 = 64 × − 1 = 64 × − 1 = 8 i
Substituting Back Now, substitute this back into the original expression: 9 − − 64 = 9 − 8 i
Final Form The complex number is 9 − 8 i , which is in the form a + bi , where a = 9 and b = − 8 .
Examples
Complex numbers are used in electrical engineering to represent alternating currents. The voltage, current, and impedance in an AC circuit can be expressed as complex numbers, making circuit analysis easier. For example, the impedance Z in an AC circuit can be represented as Z = R + j X , where R is the resistance and X is the reactance. Using complex numbers simplifies calculations involving these quantities.
