Question 13 (Essay Worth 12 points) (Order of Operations with Radicals $\mathrm{HC}$ ) A student simplified $(\sqrt[3]{27}-14 \div 2)(6-8)^{2}$ using the following steps: $(\sqrt[3]{27}-14-2)(6-8)^{2}$ \begin{tabular}{|l|l|l|} \hline Step 1: & $(3-14-2)(6-8)^{2}$ & Simplify the cube root. \\ \hline Step 2: & $(-11-2)(6-8)^{2}$ & Subtract within first parentheses. \\ \hline Step 3: & $-5.5(6-8)^{2}$ & Divide within the first parentheses. \\ \hline Step 4: & $-5.5(6-64)$ & Simplify the exponent \\ \hline Step 5: & $-5.5(-58)$ & Subtract within the parentheses. \\ \hline Step 6: & 319 & Multiply. \\ \hline \end{tabular} Part A: The student made a mistake in Step 2. Describe the mistake and explain how to correct it. (3 points) Part B: The student made a mistake in Step 4. Describe the mistake and explain how to correct it. (3 points) Part C: Show every step of your work to simplify $(\sqrt[3]{27}-14-2)(6-8)^{2}$. (6 points)
Solution
Step 1 ：For Part A, the student made a mistake in Step 2 by subtracting 2 from 14 instead of dividing 14 by 2. The correct step should be \((3-14/2)(6-8)^{2}\) which simplifies to \((3-7)(6-8)^{2}\).
Step 2 ：For Part B, the student made a mistake in Step 4 by subtracting 8 from 6 instead of squaring the result of 6-8. The correct step should be \((3-7)(-2)^{2}\) which simplifies to \(-4*4\).
Step 3 ：For Part C, the correct steps to simplify \((\sqrt[3]{27}-14/2)(6-8)^{2}\) are as follows:
Step 4 ：Step 1: \((3-14/2)(6-8)^{2}\) Simplify the cube root and division within the first parentheses.
Step 5 ：Step 2: \((3-7)(6-8)^{2}\) Subtract within first parentheses.
Step 6 ：Step 3: \(-4(6-8)^{2}\) Subtract within the first parentheses.
Step 7 ：Step 4: \(-4*(-2)^{2}\) Simplify the exponent.
Step 8 ：Step 5: \(-4*4\) Square within the parentheses.
Step 9 ：Step 6: \(-16\) Multiply.
Step 10 ：Final Answer: The final answer is \(\boxed{-16}\).
