A student simplified (cube root of 125 − 12 ÷ 2)(4 − 6)2 using the following steps:
(cube root of 125 − 12 ÷ 2)(4 − 6)2
Step 1: (5 − 12 ÷ 2)(4 − 6)2 Simplify the cube root.
Step 2: (−7 ÷ 2) (4 − 6)2 Subtract within first parentheses.
Step 3: −6.5(4 − 6)2 Divide within the first parentheses.
Step 4: −6.5(4 − 36) Simplify the exponent.
Step 5: −6.5(−32) Subtract within the parentheses.
Step 6: 208 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 125 − 12 ÷ 2)(4 − 6)2. (6 points)
Part A: The mistake in Step 2 is when the student subtracted within the parentheses. They subtracted 4 - 6 which is incorrect. To correct it, they should have subtracted 6 - 4 instead, since the order of subtraction matters.
Part B: The mistake in Step 4 is when the student simplified the exponent. They should have squared each term separately, but they squared the entire parentheses. To correct it, they should square each term individually.
Part C:
(cuberoot(125) - 12/2)(4 - 6)^2
Step 1: (5 - 12/2)(4 - 6)^2 Simplify the cube root.
Step 2: (5 - 6)(4 - 6)^2 Simplify the division within the parentheses.
Step 3: (-1)(-2)^2 Simplify the subtraction within the parentheses.
Step 4: (-1)(4) Simplify the exponent.
Step 5: -4 Multiply.
Therefore,
(cuberoot(125) - 12/2)(4 - 6)^2 simplifies to -4.
Part A: The student made a mistake in Step 2. The mistake is that they subtracted within the first parentheses incorrectly.
To correct the mistake, we need to perform the division first and then subtract within the parentheses.
Correct Step 2: (5 - (12 ÷ 2))(4 - 6)2
Part B: The student made a mistake in Step 4. The mistake is that they simplified the exponent incorrectly.
To correct the mistake, we need to correctly simplify the exponent as (4 - 6)^2 = (-2)^2 = 4.
Correct Step 4: (-7 ÷ 2)(4 - 6)^2
Part C:
Given expression: (cube root of 125 - 12 ÷ 2)(4 - 6)^2
Step 1: (5 - 12 ÷ 2)(4 - 6)^2 Simplify the cube root.
Step 2: (5 - 6)(4 - 6)^2 Subtract within the parentheses.
Step 3: (-1)(-2)^2 Simplify the exponent.
Step 4: (-1)(4) Calculate the exponent.
Step 5: -4 Multiply.
Therefore, the simplified expression is -4.
Part A: The mistake in Step 2 is in subtracting within the first parentheses. The student subtracted 12 ÷ 2 from 5, resulting in 5 - 6 = -1. However, the correct subtraction should be performed first within the parentheses.
To correct this mistake, the student should subtract 12 ÷ 2 first:
Step 2 Correction: (5 − 6) (4 − 6)2 = (-1) (4 - 6)2
Part B: The mistake in Step 4 is in simplifying the exponent. The student incorrectly calculated (4 - 6)^2 as 36.
To correct this mistake, we need to calculate the exponent correctly:
Step 4 Correction: (-1) (4 - 6)^2 = (-1) (-2)^2
Part C: To simplify (cube root of 125 − 12 ÷ 2)(4 − 6)2, let's follow the correct steps:
Step 1: Simplify the cube root.
Cube root of 125 = 5
So, the expression becomes: (5 − 12 ÷ 2)(4 − 6)2
Step 2: Subtract within first parentheses.
12 ÷ 2 = 6
So, the expression becomes: (5 − 6)(4 − 6)2
Step 3: Divide within the first parentheses.
5 - 6 = -1
So, the expression becomes: (-1)(4 − 6)2
Step 4: Simplify the exponent.
4 - 6 = -2
So, the expression becomes: (-1)(-2)^2
Step 5: Subtract within the parentheses.
-2^2 = -4
So, the expression becomes: (-1)(-4)
Step 6: Multiply.
(-1)(-4) = 4
Therefore, (cube root of 125 − 12 ÷ 2)(4 − 6)2 simplifies to 4.