Verified Answer :

  • Represent the expression as the sum of complex numbers: \((0 + 8i) + (-7 - i)\).
  • Add the real components: \(0 + (-7) = -7\).
  • Add the imaginary components: \(8i - i = 7i\).
  • Combine the results to get the final simplified complex number: \(\boxed{-7 + 7i}\).

Explanation

  1. Identifying the components of the complex numbers

To simplify the expression \(8i + (-7 - i)\), we identify this as the addition of two complex numbers. A complex number is represented in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. We can rewrite the first term as \(0 + 8i\), where the real part is \(0\) and the imaginary part is \(8\). The second term is \(-7 - 1i\), where the real part is \(-7\) and the imaginary part is \(-1\).

  1. Adding real and imaginary components

When adding complex numbers, we combine their corresponding real and imaginary parts. The sum is calculated as:
\[(a + bi) + (c + di) = (a + c) + (b + d)i\]
Applying this to our expression, we sum the real parts and the imaginary parts separately:
Real part: \(0 + (-7) = -7\)
Imaginary part: \(8i + (-i) = (8 - 1)i = 7i\)

  1. Final simplification and conclusion

Combining these results gives us the final complex number \(-7 + 7i\). Comparing this result with the given choices, it matches option C. Therefore, the simplified expression is \(\boxed{-7 + 7i}\).

Examples

Complex numbers are essential in electrical engineering for analyzing circuits, especially when dealing with alternating current (AC). For instance, when adding different voltage or current sources that operate with phase shifts, complex numbers allow engineers to efficiently calculate the total combined effect. By representing signals as complex numbers (phasors), engineers can use simple addition to find the resulting waveform, which is much faster than solving complex trigonometric differential equations.

Related Questions