Math Error: Did Jacki Solve It Right?

by Andrew McMorgan 38 views

Hey guys! Let's dive into a common math puzzle that popped up recently. Our friend Jacki was working on an expression, and we need to figure out where things might have gone a little sideways. The expression is: 23(3−1)+4(8−12)2^3(3-1)+4(8-12). Jacki's steps were:

$egin{array}{c} 2^3(3-1)+4(8-12) \ 2^3(2)+4(4) \ 8(2)+16 \ 16+16 \ 32 \ ext{Error} ext{The final answer is 32}

Now, the big question is: What was Jacki's error? We've got a couple of options to consider: A. Jacki should have simplified the exponent first. B. Jacki should have multiplied 4 and 8.

Let's break this down, and remember, in math, the order of operations is super important. It's like the unwritten rulebook that keeps everything consistent and logical. Without it, you'd get a different answer every time, and that would be chaos, right?

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we pinpoint Jacki's mistake, let's refresh our memory on the golden rule: the order of operations. You might have heard it called PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) or BODMAS (Brackets, Orders, Division and Multiplication from left to right, Addition and Subtraction from left to right). Both are essentially the same thing – a hierarchy for solving math problems.

  • Parentheses (or Brackets): Deal with anything inside these first. Simplify them as much as possible.
  • Exponents (or Orders): Next up are powers and roots.
  • Multiplication and Division: These are buddies and are done from left to right as they appear.
  • Addition and Subtraction: The final stage, also done from left to right.

This order is critical because it ensures everyone solving the same problem arrives at the exact same answer. It’s like following a recipe – if you skip steps or change the order, you won't get the delicious cake you intended!

Analyzing Jacki's Steps

Let's go through Jacki's work step-by-step and compare it to the order of operations:

Original Expression: 23(3−1)+4(8−12)2^3(3-1)+4(8-12)

Step 1: 23(2)+4(4)2^3(2)+4(4)

In this step, Jacki correctly dealt with the parentheses: (3−1)(3-1) became 22, and (8−12)(8-12) became −4-4. Wait a minute... Jacki wrote +4(4)+4(4) in the next step, which implies they changed the −4-4 to +4+4. This might be where the first subtle error occurred.

Step 2: 8(2)+168(2)+16

Here's where things get a bit more interesting. Jacki changed 232^3 to 88. This is correct because 232^3 means 2×2×2=82 \times 2 \times 2 = 8. However, they also changed 4(4)4(4) to 1616. This is also correct because 4×4=164 \times 4 = 16. But remember the potential sign error from Step 1? If (8−12)(8-12) was correctly evaluated as −4-4, then the expression should have been 4(−4)4(-4), which equals −16-16, not +16+16. So, there's a strong possibility Jacki made a sign error here by changing −4-4 to +4+4 when simplifying the parenthesis, and then proceeded to multiply 4×44 \times 4 instead of 4×−44 \times -4.

Step 3: 16+1616+16

This step involves the multiplication 8(2)8(2), which equals 1616. This part is correct based on the previous line. But if the previous line should have been 16+(−16)16 + (-16), this step would be different.

Step 4: 3232

Finally, 16+1616+16 equals 3232. This is mathematically correct based on the previous line.

Evaluating the Options

Now let's look at the choices provided:

A. Jacki should have simplified the exponent first.

Looking at Jacki's work, they did simplify the exponent (232^3 became 88) in Step 2. So, this option doesn't point to Jacki's error. They actually followed the order of operations correctly regarding the exponent.

B. Jacki should have multiplied 4 and 8.

This option is a bit tricky. The expression is 4(8−12)4(8-12). The order of operations dictates that you first simplify inside the parentheses. Jacki did this correctly in their first step, changing (8−12)(8-12) to (4)(4). However, as we noted, they seemed to have changed the sign, implying (8−12)(8-12) became +4+4 instead of −4-4. If they had skipped the parentheses simplification and jumped to multiplication, they might have done 4×8=324 \times 8 = 32 and 4×−12=−484 \times -12 = -48. This would be incorrect because parentheses must be handled first. So, while Jacki did simplify the parentheses, the result of that simplification and the subsequent multiplication seems to be where the error lies. The specific error wasn't failing to multiply 4 and 8, but rather how the multiplication 4(−4)4(-4) was handled (or mishandled, due to the sign error).

Identifying the Actual Error

Let's re-evaluate the problem correctly, following PEMDAS strictly:

  1. Parentheses:

    • (3−1)=2(3-1) = 2
    • (8−12)=−4(8-12) = -4 The expression becomes: 23(2)+4(−4)2^3(2) + 4(-4)

  2. Exponents:

    • 23=82^3 = 8 The expression becomes: 8(2)+4(−4)8(2) + 4(-4)

  3. Multiplication and Division (left to right):

    • 8(2)=168(2) = 16
    • 4(−4)=−164(-4) = -16 The expression becomes: 16+(−16)16 + (-16)

  4. Addition and Subtraction (left to right):

    • 16+(−16)=016 + (-16) = 0

So, the correct answer should be 00.

Now, let's go back to Jacki's steps and the given options.

Jacki's steps:

$eginarray}{c} 2^3(3-1)+4(8-12) \ 2^3(2)+4(4) ext{ <-- Problem here should be 4(-4) \ 8(2)+16 ext{ <-- Consequence of the previous error} \ 16+16 \ 32 ext{Error} ext{The final answer is 32}

Looking closely at Jacki's second line: 23(2)+4(4)2^3(2)+4(4). The first part, 23(2)2^3(2), is derived correctly from 23(3−1)2^3(3-1). The second part, 4(4)4(4), implies that (8−12)(8-12) was treated as 44 instead of −4-4. This is a sign error. Then, in the third line, 8(2)+168(2)+16, Jacki correctly performed the multiplication 8×2=168 \times 2 = 16 and also correctly performed 4×4=164 \times 4 = 16. The error originated in the simplification of the parentheses (8−12)(8-12) and the subsequent handling of the multiplication 4×(−4)4 \times (-4).

Let's reconsider the options in light of the correct answer being 00 and Jacki's steps:

  • A. Jacki should have simplified the exponent first. Jacki did simplify the exponent (23=82^3=8). This wasn't the error.

  • B. Jacki should have multiplied 4 and 8. This option is poorly worded. Jacki did perform multiplication involving 4, but it was 4×(8−12)4 \times (8-12) (or rather, 4×44 \times 4 in their flawed step). The error wasn't whether to multiply, but how to multiply after simplifying the parentheses correctly. If Jacki had not simplified the parentheses first, they might have incorrectly done 4×8=324 \times 8 = 32 and 4×−12=−484 \times -12 = -48. But Jacki did simplify the parentheses, albeit incorrectly with the sign. The error is more accurately described as a sign error in the parentheses simplification OR an error in performing the multiplication 4×(−4)4 \times (-4).

Given the provided options, neither perfectly describes the error. However, the most significant deviation from the correct process after the initial (flawed) parenthesis simplification is how the multiplication was handled. Jacki performed 4×4=164 \times 4 = 16 instead of the correct 4×(−4)=−164 \times (-4) = -16. This leads to 16+1616+16 instead of 16+(−16)16+(-16).

Let's re-examine the question and options one last time. The question asks for Jacki's error. Option B says