Evaluating Expressions: Leon's Math Journey

Hey everyone! Today, we're diving into a math problem where we'll dissect how Leon tackled the expression $-\frac{1}{2}(-4a-6)+a^2$ when $a=8$. This is a classic algebra problem, and it's a great way to review the order of operations and how to handle negative numbers and exponents. Let's see what Leon did and where he might have stumbled. Remember, learning from mistakes is key, so let's break this down together! We will thoroughly analyze each step of Leon's calculation to pinpoint any potential errors and understand the correct approach. This will help us not only evaluate the specific expression but also strengthen our overall understanding of algebraic manipulations.

Step-by-Step Analysis of Leon's Work

Leon started with the expression $-\frac{1}{2}(-4a-6)+a^2$ and the value $a=8$. Let's go through his steps one by one to see how he did. This kind of methodical analysis is crucial in mathematics because it allows us to identify exactly where a mistake occurred. By breaking down the problem into smaller parts, we can isolate the source of any errors and ensure that we understand the correct mathematical principles. Understanding the order of operations (PEMDAS/BODMAS) is absolutely critical to correctly evaluating these expressions.

Step 1: $-\frac{1}{2}(-4(8)-6)+8^2$

Here, Leon substitutes $a$ with $8$ in the original expression. This is a fundamental step in evaluating algebraic expressions and demonstrates a solid understanding of the problem's setup. The substitution is correctly performed, replacing every instance of 'a' with the value '8'. This initial setup is crucial for the rest of the calculation, as any error here would propagate through the following steps. Leon's work here looks accurate, and we can move on confidently knowing he has set up the problem correctly.

Step 2: $-\frac{1}{2}(-32-6)+8^2$

In this step, Leon simplifies the term inside the parentheses. He multiplies $-4$ by $8$, resulting in $-32$. This follows the order of operations, where multiplication is performed before subtraction. Again, Leon appears to be on the right track; he's correctly applying the rules of arithmetic. The next step is to further simplify the expression within the parentheses and deal with the exponent. This step is a direct application of the multiplication operation, which is a core concept in arithmetic and algebra. Maintaining accuracy in these fundamental operations is essential to obtaining the correct final answer.

Step 3: $-\frac{1}{2}(-38)+8^2$

Leon continues to simplify the expression by combining $-32$ and $-6$ to get $-38$ inside the parentheses. He has correctly performed the subtraction. Also, he's carrying the exponentiation operation ($8^2$) along. This shows that Leon understands the order of operations well so far. He is meticulously working through the expression step-by-step. The focus is to methodically reduce the complexity of the expression, ultimately leading to a single numerical value. It's a display of logical reasoning, which is a hallmark of good mathematical problem-solving skills.

Step 4: $-\frac{1}{2}(-38)+16$

In this step, Leon calculates $8^2$ as $16$. This is where the first error appears. He has incorrectly calculated $8^2$. The correct result should be $8*8=64$, not $16$. This suggests a misunderstanding of how exponents work. This error will significantly impact the final answer. Exponents, particularly squares and cubes, are frequently encountered in various mathematical applications, and understanding how to compute them accurately is very important.

Step 5: $-19+16$

Here, Leon simplifies $-\frac{1}{2}(-38)$ to $-19$. This calculation is correct; half of $38$ is $19$, and the negative sign is correctly applied. However, since the previous step had an error, this correct simplification doesn't save the outcome. If he had used 64, this step would have looked different. The focus on negative numbers and their correct handling is extremely important in algebra. Leon is demonstrating a good understanding of this part of the expression.

Step 6: -3

Finally, Leon calculates $-19+16$ to get $-3$. This calculation is correct based on the numbers he has, but due to the earlier error in calculating $8^2$, the final result is incorrect. The final answer is incorrect due to the error in the fourth step. Despite the correct arithmetic in the last steps, the incorrect value from the exponentiation operation renders the final result wrong. This highlights the impact of even a single error on the final outcome of the calculation.

Identifying the Error and Understanding the Correct Solution

The primary error lies in Step 4 where Leon incorrectly calculated $8^2$ as $16$ instead of the correct value of $64$. The correct approach involves following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Here’s how the expression should be correctly evaluated:

1. Substitute a = 8: $-\frac{1}{2}(-4(8)-6)+8^2$
2. Parentheses: $-\frac{1}{2}(-32-6)+8^2$
3. Parentheses: $-\frac{1}{2}(-38)+8^2$
4. Exponents: $-\frac{1}{2}(-38)+64$
5. Multiplication: $19+64$
6. Addition: $83$

Therefore, the correct answer is $83$, not $-3$. By carefully adhering to the order of operations and double-checking each step, we can avoid the types of errors Leon encountered. Understanding this is key to excelling in algebra and many other areas of mathematics. The correct evaluation demonstrates a systematic approach to mathematical problems, which is critical to success. This method will help in the accurate computation of various mathematical expressions.

Conclusion and Key Takeaways

Leon's attempt to evaluate the expression provides a valuable lesson in the importance of the order of operations and careful attention to detail. While Leon demonstrated a good grasp of the initial substitution and simplification steps, the miscalculation of $8^2$ led to an incorrect final answer. This highlights the necessity of paying close attention to the correct application of mathematical rules, particularly exponents. By recognizing and correcting these types of errors, we can strengthen our problem-solving skills and develop a deeper understanding of algebraic principles. This process is essential for mastering more complex mathematical concepts later on. Always double-check your work, and don't be afraid to break down problems into smaller steps! Keep practicing, guys, and you'll get it!