Simplifying Math Expressions: A Step-by-Step Guide

Oct 28, 2025 by ADMIN 51 views

$Simplifying the Expression: $|2-5|-(6 \\div 2+1)^2$$

Hey guys! Let's break down how to simplify this mathematical expression: $|2-5|-(6 \div 2+1)^2$ . It might look a bit intimidating at first, but don't worry, we'll tackle it step by step. Understanding the order of operations is key here, and we'll use PEMDAS/BODMAS as our guide. So, let's dive in and make math a little less scary!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we jump into simplifying our expression, it’s super important to understand the order of operations. This is often remembered by the acronyms 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, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Either way, the order is the same!

Think of it like this: we're following a set of rules to make sure we all get the same answer. If we didn't have these rules, things could get pretty confusing! So, when you see an expression, always start with what’s inside the parentheses or brackets. Then, handle any exponents or orders. After that, do multiplication and division from left to right, and finally, addition and subtraction, also from left to right. Mastering this order is absolutely crucial for simplifying any mathematical expression correctly. It’s the foundation for more complex math problems, so it’s worth taking the time to really understand it. Let's keep this order in mind as we tackle our main problem, and you'll see how smoothly it all comes together!

Step-by-Step Simplification

Let's simplify the expression $|2-5|-(6 \div 2+1)^2$ step-by-step, making sure we follow the order of operations. First, we'll deal with the absolute value and the parentheses. Remember, the absolute value of a number is its distance from zero, so it's always positive.

1. Simplify Inside the Absolute Value and Parentheses

Absolute Value: $|2-5|$ becomes $|-3|$ , which simplifies to $3$ . Think of it as how far -3 is from 0 on a number line – it’s 3 units away. So, we've taken care of the absolute value part.
Parentheses: Next up, we tackle $(6 \div 2+1)$ . Inside the parentheses, we follow the order of operations again. Division comes before addition, so we first calculate $6 \div 2$ , which equals $3$ . Then, we add $1$ , so $3 + 1 = 4$ . Now we've simplified the parentheses to $4$ .

At this point, our expression looks like this: $3 - (4)^2$ . We're making good progress! The key is to break it down bit by bit, focusing on one operation at a time. This makes the whole process much less daunting and helps prevent errors. Next, we'll handle the exponent.

2. Handle the Exponent

Now, let's deal with the exponent. We have $(4)^2$ , which means $4$ raised to the power of $2$ , or $4$ multiplied by itself. So, $4^2 = 4 * 4 = 16$ . Exponents tell us how many times to multiply a number by itself, and in this case, it's twice.

Our expression now looks even simpler: $3 - 16$ . We're in the home stretch! We've taken care of the absolute value, the parentheses, and the exponent. All that's left is one simple subtraction.

3. Perform the Subtraction

Finally, we subtract. We have $3 - 16$ . When we subtract a larger number from a smaller number, the result will be negative. In this case, $3 - 16 = -13$ . Think of it like starting at 3 on a number line and moving 16 places to the left – you'll end up at -13.

So, after all these steps, we've simplified the original expression $|2-5|-(6 \div 2+1)^2$ to $-13$ . That's our final answer! See how breaking it down and following the order of operations made it manageable? Let's recap the whole process to make sure we've got it solid.

Recap of the Simplification Process

Let's quickly recap the steps we took to simplify the expression $|2-5|-(6 \div 2+1)^2$ . This will help solidify our understanding and make sure we didn't miss anything along the way. Remember, consistency and careful attention to detail are your best friends in math!

Simplified the Absolute Value: We started by simplifying $|2-5|$ , which became $|-3|$ , and then $3$ . Absolute values always give us the positive distance from zero.
Simplified Inside the Parentheses: Next, we tackled $(6 \div 2+1)$ . Following the order of operations inside the parentheses, we divided $6$ by $2$ to get $3$ , and then added $1$ to get $4$ .
Handled the Exponent: We then dealt with the exponent, calculating $4^2$ , which equals $16$ . Exponents are a crucial part of many mathematical expressions.
Performed the Subtraction: Finally, we subtracted $16$ from $3$ , resulting in $-13$ . This gave us our final simplified answer.

So, there you have it! By following the order of operations (PEMDAS/BODMAS) and breaking the problem down into smaller, manageable steps, we successfully simplified the expression. This approach works for all sorts of mathematical problems, so keep practicing, and you'll become a simplification pro in no time! Remember, math is like building blocks – each step builds on the one before, and understanding the fundamentals is key to tackling more complex challenges. Now, let's address the answer choices provided and see which one matches our result.

Matching the Answer Choice

Okay, so we've diligently worked through the simplification process and arrived at the answer $-13$ . Now, let’s circle back to the original question's answer choices to make sure we select the correct one. This is a crucial step in any math problem – always double-check that your calculated answer matches one of the options provided. It's easy to make a small mistake along the way, so verifying your result against the choices is a smart move.

The answer choices given were:

A. -7
B. -19
C. -1
D. -13

Looking at these options, we can clearly see that our calculated answer, $-13$ , matches option D. Hooray! We got it right! This final step of matching your solution to the provided choices is a great way to ensure accuracy and build confidence in your problem-solving skills. It also helps reinforce the process in your mind, making you even better prepared for similar problems in the future. Remember, math is all about practice, so the more you work through problems like this, the more comfortable and skilled you'll become. Let's wrap things up with a final thought on why mastering these simplification skills is so important.

Why Simplification Skills Matter

Guys, mastering simplification skills is super important in math and beyond! It's not just about getting the right answer to this one problem; it's about building a foundation for more advanced math concepts and developing problem-solving skills that you can use in all areas of life. Think of it like this: simplification is like learning the alphabet of mathematics. Once you've got the basics down, you can start putting things together to form more complex ideas and solve more challenging problems.

When you can confidently simplify expressions, you're better equipped to tackle algebra, calculus, and even real-world problems that involve numbers and calculations. These skills are essential for many fields, from science and engineering to finance and technology. Plus, the ability to break down a complex problem into smaller, more manageable steps is a valuable skill in itself. It teaches you to think logically, stay organized, and approach challenges with a clear and methodical mindset. So, keep practicing your simplification skills, and you'll be amazed at how far they take you! You'll not only excel in math class but also develop a powerful set of tools for navigating the world around you. And remember, every problem you solve is a step closer to mastering these crucial skills. Keep up the great work!