Which pair of complex factors results in a real-number product?
The goal is to determine which of the given pairs of complex numbers results in a real-number product. A complex number \(z = x + yi\) is considered a real number if its imaginary part \(y\) is equal to \(0\). We can test each option by calculating the product of the given complex factors.
We evaluate each option individually:
The product is \(-(8+20i)^2 = -(64 + 320i - 400) = -(-336 + 320i) = 336 - 320i\). Since the imaginary part is \(-320 eq 0\), this product is not a real number.
The product is \(-225i\). Since the imaginary part is \(-225 eq 0\), this product is not a real number.
The product is \(3i - 9i^2 = 3i + 9 = 9 + 3i\). Since the imaginary part is \(3 eq 0\), this product is not a real number.
This is a product of a complex number and its conjugate. Using the difference of squares identity \((x+yi)(x-yi) = x^2 + y^2\), we have \(4^2 - (7i)^2 = 16 - 49(-1) = 16 + 49 = 65\). Since the imaginary part is \(0\), this product is a real number.
Comparing the results, only option d results in a purely real product. Therefore, the pair \((4+7i)(4-7i)\) is the only one that yields a real number. The final answer is \(\boxed{d}\).
In electrical engineering, calculating impedance often involves multiplying complex numbers. When determining the power dissipated by a component, we look for real-valued components of the result. For instance, when multiplying an impedance \(Z = R + jX\) by its conjugate \(Z^* = R - jX\), the result is \(|Z|^2 = R^2 + X^2\), which is a purely real number representing the magnitude squared. This concept is fundamental for simplifying complex circuit analysis into real-world power calculations.