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 that the student subtracted within the first parentheses when they should have multiplied. To correct it, the expression should be multiplied instead of subtracted.
Part B: The mistake in Step 4 is that the student simplified the exponent incorrectly. The exponent should be squared, not subtracted. To correct it, the expression should be squared instead of subtracted.
Part C:
(cube root of 125 − 12 ÷ 2)(4 − 6)2
Step 1: (5 - 12 ÷ 2)(4 - 6)2
Step 2: (5 - 6)(4 - 6)2
Step 3: (-1)(4 - 6)2
Step 4: (-1)(-2)2
Step 5: (-1)(4)
Step 6: -4
Part A: The mistake in Step 2 is that the student subtracted within the parentheses after dividing within the first parentheses. The correct order of operations is to simplify within the parentheses first and then perform the division.
To correct the mistake, the student should perform the division before subtracting:
Step 2 Correction:
(−7 ÷ 2) (4 − 6)2
(-3.5)(4 - 6)2
Part B: The mistake in Step 4 is that the student simplified the exponent incorrectly. The exponent should be applied to both terms in the parentheses, not just the second term.
To correct the mistake, the student should raise both terms inside the parentheses to the power of 2:
Step 4 Correction:
(-3.5)(4 - 6)^2
(-3.5)(-2)^2
Part C: To simplify (cube root of 125 − 12 ÷ 2)(4 − 6)^2, follow these steps:
Step 1: Start with the given expression.
(cube root of 125 − 12 ÷ 2)(4 − 6)^2
Step 2: Simplify the cube root.
(5 − 12 ÷ 2)(4 − 6)^2
Step 3: Perform the division within the first parentheses.
(5 - 6)(4 - 6)^2
Step 4: Apply the exponent to both terms inside the second parentheses.
(5 - 6)(-2)^2
Step 5: Simplify the exponent by squaring the value inside the parentheses.
(5 - 6)(4)
Step 6: Perform the subtraction within the first parentheses.
(-1)(4)
Step 7: Multiply the two terms together to get the final result.
-4
Part A: The mistake in Step 2 is that the student incorrectly subtracted within the first set of parentheses. Instead of getting (-7 ÷ 2), the correct value should be (5 - 6 ÷ 2). To correct the mistake, we need to follow the correct order of operations, which is parentheses first, then division.
Step 2 Correction: (5 - 6 ÷ 2) (4 - 6)2
Part B: The mistake in Step 4 is that the student incorrectly simplified the exponent. Instead of getting (4 - 36) as the exponent, the correct value should be (4 - 6) as the base, and the exponent should remain as 2. To correct the mistake, we need to clearly indicate that the exponent applies only to the base, not to the subtracted value.
Step 4 Correction: -6.5(4 - 6)2
Part C: To simplify (cube root of 125 - 12 ÷ 2)(4 - 6)2, we'll follow the correct sequence of steps:
Step 1: Start by evaluating the cube root of 125, which is 5, and simplify the division 12 ÷ 2, which is 6.
(5 - 6)(4 - 6)2
Step 2: Subtract within the parentheses (5 - 6) and (4 - 6).
(-1)(-2)2
Step 3: Simplify the exponent, which in this case is 2.
-1(-2)(-2)
Step 4: Multiply the values inside the parentheses.
-1 * -2 * -2
Step 5: Perform the multiplications.
-1 * -2 * -2 = -2 * -2 = 4
Therefore, the simplified expression is 4.