Which of the following is $|8-6 i|$?
-10
$2 \sqrt{7}$
$2 i \sqrt{7}$
10
Answer (1)
We are asked to find the modulus of the complex number 8 − 6 i .
The modulus of a complex number a + bi is given by a 2 + b 2 .
Substituting a = 8 and b = − 6 into the formula, we get ∣8 − 6 i ∣ = 8 2 + ( − 6 ) 2 = 64 + 36 = 100 .
Therefore, ∣8 − 6 i ∣ = 10 .
Explanation
Understanding the Problem We are asked to find the modulus (or absolute value) of the complex number 8 − 6 i . The modulus of a complex number a + bi is given by the formula a 2 + b 2 . In this case, a = 8 and b = − 6 .
Applying the Formula To find the modulus of 8 − 6 i , we substitute a = 8 and b = − 6 into the formula ∣ a + bi ∣ = a 2 + b 2 . This gives us ∣8 − 6 i ∣ = 8 2 + ( − 6 ) 2 .
Calculating the Squares Now, we calculate the squares: 8 2 = 64 and ( − 6 ) 2 = 36 . So, we have ∣8 − 6 i ∣ = 64 + 36 = 100 .
Finding the Square Root Finally, we take the square root: 100 = 10 . Therefore, the modulus of the complex number 8 − 6 i is 10.
Conclusion The modulus of the complex number 8 − 6 i is 10. Therefore, the correct answer is 10.
Examples
Complex numbers are used in electrical engineering to analyze alternating current (AC) circuits. The impedance of a circuit, which is the opposition to the flow of current, can be represented as a complex number. The modulus of the impedance gives the magnitude of the impedance, which is a real number that represents the total opposition to current flow. For example, if the impedance of a circuit is 8 − 6 i ohms, then the magnitude of the impedance is ∣8 − 6 i ∣ = 10 ohms, indicating the total opposition to current flow in the circuit.
