Final answer:
To simplify (4 - 8i)(2 - 7i), one must apply the distributive property and combine like terms, correctly applying i² = -1. The final answer is -48 - 44i, which is the second option provided.
Step-by-step explanation:
The student is asking to simplify the product of two complex numbers: (4 - 8i)(2 - 7i). This simplification is a matter of applying the distributive property, also known as the FOIL method for binomials, and then combining like terms.
- First, multiply the real parts: 4 * 2 = 8.
- Second, multiply the outer parts: 4 * -7i = -28i.
- Third, multiply the inner parts: -8i * 2 = -16i.
- Lastly, multiply the imaginary parts: -8i * -7i = 56i² (since i² = -1, this becomes 56 * -1 = -56).
Combining these products gives us 8 - 28i - 16i - 56. Now add the like terms (the imaginary terms -28i and -16i combine to -44i), and this simplifies to 8 - 56 - 44i. The final step is to combine the real numbers (8 and -56), which results in -48 - 44i. To check if the answer is reasonable, we ensure that all terms have been correctly combined and that the property i² = -1 has been correctly applied. The correct answer simplifies down to -48 - 44i, which corresponds to option 2).