Question

Question 13 (Essay Worth 12 points)

(Order of Operations with Radicals HC)

A student simplified (cube root of 27 − 14 ÷ 2)(6 − 8)2 using the following steps:

(cube root of 27 − 14 ÷ 2)(6 − 8)2


Step 1: (3 − 14 ÷ 2)(6 − 8)2 Simplify the cube root.
Step 2: (−11 ÷ 2) (6 − 8)2 Subtract within first parentheses.
Step 3: −5.5(6 − 8)2 Divide within the first parentheses.
Step 4: −5.5(6 − 64) Simplify the exponent.
Step 5: −5.5(−58) Subtract within the parentheses.
Step 6: 319 Multiply.


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 (cube root of 27 − 14 ÷ 2)(6 − 8)2. (6 points)

Answer 1

Part A: The mistake in Step 2 is subtracting within the first parentheses before dividing. The correct order of operations is to perform division before subtraction. The correct step should be:

\[ (−11 ÷ 2) (6 − 8)2 \]

\[ (-5.5) \cdot (6 - 8)^2 \]

\[ -5.5 \cdot (-2)^2 \]

\[ -5.5 \cdot 4 \]

\[ -22 \]

Part B: The mistake in Step 4 is not simplifying the exponent correctly. The correct simplification of the exponent should be:

\[ -5.5(6 - 8)^2 \]

\[ -5.5(-2)^2 \]

\[ -5.5 \cdot 4 \]

\[ -22 \]

Part C: Simplifying \( (\sqrt[3]{27} - \frac{14}{2})(6 - 8)^2 \):

\[ (3 - \frac{14}{2})(6 - 8)^2 \]

\[ (-11)(-2)^2 \]

\[ -11 \cdot 4 \]

\[ -44 \]