Verified Answer :

  • Use the binomial square formula \((A-B)^2 = A^2 - 2AB + B^2\) where \(A=8\) and \(B=5i\).
  • Substitute the values: \(8^2 - 2(8)(5i) + (5i)^2 = 64 - 80i + 25i^2\).
  • Simplify using \(i^2 = -1\): \(64 - 80i - 25 = 39 - 80i\).
  • The correct option is \(\boxed{c}\).

Explanation

  1. Analyzing the Problem

To solve the expression \((8-5i)^2\), we recognize it as the square of a binomial. The general formula for the square of a difference is \((A-B)^2 = A^2 - 2AB + B^2\). In our case, \(A = 8\) and \(B = 5i\).

  1. Expanding the Binomial

We apply the expansion formula \((A-B)^2 = A^2 - 2AB + B^2\) using \(A = 8\) and \(B = 5i\):
\[(8-5i)^2 = 8^2 - 2(8)(5i) + (5i)^2\]
First, calculate the individual terms:

  • \(8^2 = 64\)
  • \(2(8)(5i) = 80i\)
  • \((5i)^2 = 25i^2\)

Since the imaginary unit \(i\) is defined by \(i^2 = -1\), we substitute this into the last term:

  • \(25i^2 = 25(-1) = -25\)
  1. Final Calculation and Result

Now, combine the simplified terms from our expansion:
\[(8-5i)^2 = 64 - 80i - 25\]
Group the real parts together:
\[(64 - 25) - 80i = 39 - 80i\]
The final result is \(39 - 80i\), which corresponds to option c.

Examples

Complex numbers, like \((8-5i)^2\), appear frequently in electrical engineering when analyzing alternating current (AC) circuits. They help represent the relationship between voltage, current, and impedance (resistance and reactance). For instance, when calculating the power in a circuit, engineers often square complex expressions representing impedance to determine total power consumption, helping ensure that electronic devices operate safely and efficiently within their design limits.

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