Math Error Analysis: Jacki's Expression Mishap

by Omar Yusuf 47 views

Hey guys! Ever made a math mistake that seemed to slip right past you? We've all been there! Today, we're diving into a problem Jacki tackled and figuring out where things went a little sideways. Let's break down the problem, Jacki's steps, and pinpoint exactly where the error occurred. This is a fantastic way to sharpen our own math skills and become expert error-detectors!

The Problem: Unraveling the Expression

Jacki was faced with the following expression:

2^3(3-1) + 4(8-12)

This looks like a classic order of operations problem, and we know PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is our trusty guide. Understanding the order of operations is absolutely crucial for simplifying any mathematical expression correctly. It's like having a roadmap for solving the problem, ensuring we tackle each operation in the right sequence. Let's keep PEMDAS in mind as we analyze Jacki's solution.

Jacki's Steps: A Walkthrough

Let's follow Jacki's journey through the problem. Here's how she broke it down:

  1. Step 1: 2^3(3-1) + 4(8-12)
  2. Step 2: 2^3(2) + 4(4) (Oops! This is where a potential error might be hiding. Let's keep going and see if we can catch it in the act.)
  3. Step 3: 8(2) + 16
  4. Step 4: 16 + 16
  5. Step 5: 32

At first glance, it seems like Jacki followed the steps logically. But, like any good math detectives, we need to scrutinize each step to be absolutely sure. Did Jacki truly follow PEMDAS to the letter? Or did a sneaky error creep in along the way? Let's put on our detective hats and dig deeper!

Spotting the Error: The Devil's in the Details

Okay, time to put on our critical thinking caps! Let's go back to Jacki's steps and meticulously check each one against the order of operations (PEMDAS). Remember, parentheses first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

  • Step 1: 2^3(3-1) + 4(8-12) - This is the original expression. No errors here.
  • Step 2: 2^3(2) + 4(4) - Wait a minute! Did Jacki make a mistake within the second set of parentheses? (8-12) should equal -4, not 4. This is our prime suspect! It looks like Jacki might have overlooked the negative sign. This is a crucial error because it changes the entire outcome of the problem. We need to make sure we don't make this mistake ourselves. Remember, paying close attention to signs (+ and -) is a key to success in math!
  • Step 3: 8(2) + 16 - This step is a consequence of the error in Step 2. Since the previous calculation was incorrect, this step is also heading down the wrong path.
  • Step 4: 16 + 16 - Again, this step is based on the previous incorrect result.
  • Step 5: 32 - The final answer is incorrect due to the initial error in handling the negative sign within the parentheses.

The Verdict: The error lies in Step 2. Jacki incorrectly simplified (8-12) as 4 instead of -4. This seemingly small mistake had a ripple effect, leading to an incorrect final answer. We've solved the mystery! The key takeaway here is the importance of paying close attention to signs and the order of operations.

Why the Error Matters: Understanding the Impact

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