Question

multiply.
\frac{7}{2x - 16} \cdot \frac{4x - 32}{5}
simplify your answer as much as possible.

Explanation:

Step1: Factor numerators and denominators

Factor \(2x - 16\) as \(2(x - 16)\)? Wait, no, \(2x - 16 = 2(x - 8)\), and \(4x - 32 = 4(x - 8)\). So the expression becomes \(\frac{7}{2(x - 8)} \cdot \frac{4(x - 8)}{5}\)

Step2: Cancel common factors

The \((x - 8)\) terms cancel out, and \(4\) and \(2\) can be simplified. \(4\div2 = 2\). So we have \(\frac{7}{2} \cdot \frac{4}{5}\) (after canceling \((x - 8)\))? Wait, no, let's do it step by step. After factoring, it's \(\frac{7}{2(x - 8)} \times \frac{4(x - 8)}{5}\). The \((x - 8)\) cancels, \(4\) and \(2\) simplify: \(4/2 = 2\). So now we have \(\frac{7 \times 2}{5}\)? Wait, no, wait: \(\frac{7}{2(x - 8)} \times \frac{4(x - 8)}{5}\) = \(\frac{7 \times 4(x - 8)}{2(x - 8) \times 5}\). Cancel \((x - 8)\): \(\frac{7 \times 4}{2 \times 5}\). Then \(4/2 = 2\), so \(\frac{7 \times 2}{5} = \frac{14}{5}\)? Wait, no, wait: 7 times 4 is 28, 2 times 5 is 10, then 28/10 simplifies to 14/5? Wait, no, 4 divided by 2 is 2, so 7 times 2 is 14, over 5. Yes. Wait, let's check again. Original expression: \(\frac{7}{2x - 16} \times \frac{4x - 32}{5}\). Factor denominators and numerators: \(2x - 16 = 2(x - 8)\), \(4x - 32 = 4(x - 8)\). So substitute: \(\frac{7}{2(x - 8)} \times \frac{4(x - 8)}{5}\). Now, multiply numerators: 7 4(x - 8) = 28(x - 8). Multiply denominators: 2(x - 8) 5 = 10(x - 8). Now, cancel (x - 8) from numerator and denominator: 28/10. Simplify 28/10 by dividing numerator and denominator by 2: 14/5. Wait, but 14/5 is 2.8, but let's check the steps again. Wait, 7 times 4 is 28, 2 times 5 is 10, 28 divided by 10 is 14/5. Yes. So the simplified form is 14/5? Wait, no, wait: 7(4(x - 8)) = 28(x - 8); 2(x - 8)5 = 10(x - 8). Cancel (x - 8): 28/10 = 14/5. Yes. So the final answer is 14/5? Wait, but let's do it again. Step 1: Factor 2x - 16 = 2(x - 8), 4x - 32 = 4(x - 8). Step 2: Rewrite the expression as (7 / [2(x - 8)]) (4(x - 8) / 5). Step 3: Multiply numerators: 7 4(x - 8) = 28(x - 8). Multiply denominators: 2(x - 8) * 5 = 10(x - 8). Step 4: Cancel (x - 8) from numerator and denominator: 28/10. Step 5: Simplify 28/10 by dividing numerator and denominator by 2: 14/5. Yes, that's correct.

Answer:

\(\frac{14}{5}\)