Question
Which multiplication expression is equivalent to \( \begin{array}{l}\frac{x+8}{x^{2}} \div \frac{2 x+16}{2 x^{2}} ? \\ \frac{x^{2}}{x+8} \cdot \frac{2 x^{2}}{2 x+16} \\ \frac{x+8}{x^{2}} \cdot \frac{2 x+16}{2 x^{2}} \\ \frac{x+8}{x^{2}} \cdot \frac{2 x^{2}}{2 x+16}\end{array} \)
Ask by Moran Fleming. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The correct multiplication expression is \( \frac{x+8}{x^2} \cdot \frac{2x^2}{2x+16} \).
Solution
1. The original expression is
\[
\frac{x+8}{x^2} \div \frac{2x+16}{2x^2}.
\]
2. Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the expression as
\[
\frac{x+8}{x^2} \cdot \frac{2x^2}{2x+16}.
\]
3. Comparing this result with the provided choices, we see that it matches
\[
\frac{x+8}{x^2} \cdot \frac{2x^2}{2x+16}.
\]
4. Hence, the correct multiplication expression is
\[
\frac{x+8}{x^2} \cdot \frac{2x^2}{2x+16}.
\]
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Simplify this solution Bonus Knowledge
To simplify the expression \( \frac{x+8}{x^{2}} \div \frac{2 x+16}{2 x^{2}} \), you can rewrite the division as multiplication by the reciprocal. Thus, it becomes: \[ \frac{x+8}{x^{2}} \cdot \frac{2 x^{2}}{2 x+16} \] Noting that \( 2x + 16 \) can be factored as \( 2(x + 8) \), the expression simplifies to: \[ \frac{x+8}{x^{2}} \cdot \frac{2 x^{2}}{2(x + 8)} \] The \( x + 8 \) terms cancel each other out, leading us to: \[ \frac{1}{x^{2}} \cdot \frac{2 x^{2}}{2} = 1 \] Therefore, the equivalent expression is: \[ \frac{x+8}{x^{2}} \cdot \frac{2 x^{2}}{2 x + 16} \]
