Easy Math: Evaluate $6+(7 imes 8)-2-5$

Welcome, math enthusiasts and curious minds! Today, we're going to dive into a fun mathematical expression and break it down step-by-step. Evaluating expressions is a fundamental skill in mathematics, and mastering it will unlock a deeper understanding of more complex concepts. The expression we'll be tackling is $6+(7 imes 8)-2-5$. Don't let the combination of numbers and operations intimidate you; with a systematic approach, we'll unravel it with ease. We'll be adhering to the order of operations, a set of rules that ensures everyone arrives at the same correct answer for any given mathematical expression. Think of it as a universal language for numbers! The order of operations is often remembered by the acronym PEMDAS or BODMAS. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS stands for Brackets, Orders (powers and square roots, etc.), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Both acronyms guide us through the hierarchy of calculations, ensuring consistency and accuracy in our mathematical endeavors. Understanding this hierarchy is crucial, as performing operations in the wrong order can lead to significantly different, and incorrect, results. For instance, if we were to simply go from left to right without considering the order of operations, we might calculate $7 imes 8$ as $56$ (correctly in this case, but let's imagine a scenario where it wasn't), but then if we added $6+56$ to get $62$, and then subtracted $2$ to get $60$, and then subtracted $5$ to get $55$, and finally tried to add the $7 imes 8$ part after everything else, the answer would be wildly different. It’s a bit like following a recipe; you wouldn't put the cake in the oven before mixing the ingredients, would you? Similarly, in mathematics, there’s a specific sequence to follow. We'll walk through each step, explaining the reasoning behind it, so you can confidently tackle similar problems on your own. Whether you're a student working through homework or simply someone looking to sharpen your mathematical skills, this breakdown will be incredibly beneficial. Let's get started on evaluating our expression and discover the final numerical value!

Understanding the Order of Operations (PEMDAS/BODMAS)

To effectively evaluate the expression $6+(7 imes 8)-2-5$, we must first understand and apply the order of operations. This is a universal convention in mathematics that dictates the sequence in which different mathematical operations should be performed to ensure a single, consistent, and correct answer. You might have heard of the acronyms PEMDAS or BODMAS. Let's break down what each letter represents:

• Parentheses (or Brackets): These are grouping symbols that indicate the operations within them must be performed first. If there are nested parentheses, we start with the innermost set.
• Exponents (or Orders): This refers to powers and roots. For example, $2^3$ (2 cubed) or $\sqrt{9}$ (the square root of 9).
• Multiplication and Division: These operations have equal priority and are performed from left to right as they appear in the expression.
• Addition and Subtraction: These operations also have equal priority and are performed from left to right as they appear in the expression.

In our expression, $6+(7 imes 8)-2-5$, we can see parentheses. This immediately tells us that the operation inside the parentheses needs to be our first focus. The operation inside is $7 imes 8$. By following the order of operations, we isolate this part of the calculation. This principle is incredibly important because if we were to ignore it, we could arrive at drastically different answers. Imagine performing addition before multiplication; the result would be quite altered. For example, if we did $6+7$ first, we'd get $13$, and then $13 imes 8 = 104$, which is far from the correct answer. The strict adherence to PEMDAS/BODMAS ensures that all mathematicians, regardless of their location or background, will reach the identical solution when evaluating the same expression. It's the bedrock of mathematical consistency and predictability. Understanding this hierarchy isn't just about memorizing an acronym; it's about grasping the logical flow of mathematical problem-solving. It establishes a clear pathway to deconstruct complex calculations into manageable steps, making even daunting expressions approachable. So, let's put these rules into practice and tackle our expression systematically.

Step-by-Step Evaluation of the Expression

Now that we have a firm grasp on the order of operations, let's apply it to evaluate the expression $6+(7 imes 8)-2-5$. We will proceed through each step methodically, ensuring accuracy and clarity. Remember, the goal is to simplify the expression by performing one operation at a time, following the PEMDAS/BODMAS hierarchy.

Step 1: Operations within Parentheses

The first step according to PEMDAS/BODMAS is to address any operations enclosed in parentheses. In our expression, $6+(7 imes 8)-2-5$, we find $(7 imes 8)$. This multiplication must be performed before any addition or subtraction outside the parentheses.

• Calculate the product: $7 imes 8 = 56$.

Now, we can substitute this result back into the original expression. The expression becomes:

• $6 + 56 - 2 - 5$

This is a crucial first step, as it simplifies the expression significantly and isolates the remaining operations. By resolving the part within the parentheses, we've reduced the complexity and are one step closer to our final answer. It's like clearing the most immediate obstacle before moving on to the broader path. This initial simplification makes the subsequent steps much more straightforward and less prone to error.

Step 2: Addition and Subtraction (from Left to Right)

After handling the parentheses, we move on to addition and subtraction. These operations have equal priority, so we perform them from left to right as they appear in the expression. Our current expression is $6 + 56 - 2 - 5$.

• Perform the first operation from the left: $6 + 56$.
  • $6 + 56 = 62$.

Now, our expression is updated to:

• $62 - 2 - 5$

Next, we continue moving from left to right and perform the next subtraction:

• Perform the next operation: $62 - 2$.
  • $62 - 2 = 60$.

Our expression is now:

• $60 - 5$

Finally, we perform the last subtraction:

• Perform the final operation: $60 - 5$.
  • $60 - 5 = 55$.

And there you have it! After systematically applying the order of operations, we have arrived at the final value of the expression.

The Final Answer

By carefully following the established rules of mathematics, specifically the order of operations (PEMDAS/BODMAS), we have successfully evaluated the expression $6+(7 imes 8)-2-5$. Let's recap the journey:

1. Parentheses: We first tackled the operation within the parentheses, $7 imes 8$, which equals $56$. The expression then simplified to $6 + 56 - 2 - 5$.
2. Addition and Subtraction (Left to Right): We proceeded with the addition: $6 + 56 = 62$. The expression became $62 - 2 - 5$.
3. Next, we performed the subtraction from left to right: $62 - 2 = 60$. The expression further simplified to $60 - 5$.
4. Finally, we completed the last subtraction: $60 - 5 = 55$.

Therefore, the evaluated value of the expression $6+(7 imes 8)-2-5$ is 55.

Mastering these fundamental skills is key to building a strong foundation in mathematics. It's not just about getting the right answer; it's about understanding the logic and the systematic process that leads to it. This methodical approach ensures accuracy and builds confidence for tackling more complex mathematical challenges in the future.

If you're looking to explore more about the order of operations and practice additional problems, a great resource is the Khan Academy Mathematics section. They offer a wealth of free resources, explanations, and practice exercises that can further solidify your understanding.