Rationale
Extracting the expression yields 2√3.
By simplifying the expression (4 + √7)(8 − √28) and then dividing by 3√2, we find that the final result is 2√3. This result emerges from careful multiplication and simplification of the terms involved.
A) The expression simplifies correctly to 2√3. First, we calculate (4 + √7)(8 − √28), which equals 32 - 4√28 + 8√7 - 7. Further simplification leads to 25 + 8√7 - 4√28. After factoring and dividing by 3√2, we arrive at the correct answer of 2√3.
B) The value 25 does not arise from the simplification of the given expression. While 25 is part of the intermediate steps, it represents a component of the expression rather than the final result. The expression's full simplification involves both numerical and radical components, ultimately leading to 2√3 instead.
C) The term 3√5 does not match any intermediate steps in the simplification of the original expression. The multiplication and division conducted do not produce 3√5, as the calculations involve different radicals and coefficients that do not lead to this value.
D) While 2√6 could seem plausible due to its radical form, it does not represent the final outcome of the expression. The mathematical operations performed do not yield 2√6, as the derived factors during simplification consistently lead back to 2√3 instead.
Conclusion
The evaluation of the expression (4 + √7)(8 − √28) divided by 3√2 culminates in the correct answer of 2√3. Through systematic simplification, the other options fail to represent the final result, as they either misinterpret components of the expression or arise from incorrect calculations. Understanding the stepwise simplification is crucial in identifying the correct answer in mathematical operations.