Final answer:
The student needs to understand the relationship between multiplication and division with fractions, specifically the concept of multiplying by the reciprocal. The correct expression resulting from the given division problem, assuming no typos, would be (x + 8/x²) * (x + 16). None of the provided options match this result.
Step-by-step explanation:
To solve the given problem, we need to understand the rules of fractions and how they relate to multiplication and division. In general, dividing by a fraction is equivalent to multiplying by its reciprocal. Similarly, multiplying by a fraction is akin to dividing by its reciprocal. The task requires us to identify an expression equivalent to (x 8/ x²) divided by (2x 16/2x²). Using these principles, we will convert the division into multiplication by the reciprocal.
First, let's simplify the given expressions:
- We write the division as multiplication by the reciprocal: (x + 8/x²) * (2x²/2x + 16).
- Next, we simplify the fractions where possible, noting that 2x²/2x simplifies to x.
Thus, the simplified multiplication expression is (x + 8/x²) * (x + 16), which does not match any of the options provided, suggesting a potential typo in the original question.
Important Rules to Remember:
- When multiplying or dividing both sides of an equation by the same number, the equality is maintained.
- Multiplication or division should apply to every term, which sometimes requires the use of parentheses to enclose expressions with multiple terms.
- The sign rules for multiplication and division indicate that two positives or two negatives result in a positive, while a positive and a negative result in a negative.