Solve 1/2[2(2^3+6)]-6: A Step-by-Step Guide

Introduction

Hey guys! Today, we're diving deep into a mathematical problem that might seem a bit intimidating at first glance, but trust me, we'll break it down step by step until it's as clear as day. Our mission is to solve the expression 1/2[2(2^3+6)]-6. Math can be like a puzzle, and this one is definitely worth solving. Whether you're brushing up on your math skills, helping with homework, or just enjoy a good brain teaser, you're in the right place. We'll use the order of operations, also known as PEMDAS or BODMAS, to guide us through this. This order – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) – is our trusty map to navigate the world of mathematical expressions. So, grab your pencils, and let's get started!

Our journey today is more than just finding the right answer; it's about understanding the process. Each step in solving this equation is a building block, adding to your mathematical knowledge and boosting your confidence. We'll take our time, explain every move, and ensure you're not just memorizing steps, but truly grasping the hows and whys behind them. This approach will not only help you with this particular problem but will also equip you to tackle other mathematical challenges with ease. Remember, math is a language, and once you understand its rules, you can read and speak it fluently. So, let’s embark on this mathematical adventure together and unlock the solution to 1/2[2(2^3+6)]-6.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we even think about touching the numbers, let's talk about our golden rule: the order of operations. You might have heard of it as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). They both mean the same thing – a set of rules that tell us the exact order in which we should perform mathematical operations. Imagine trying to build a house without a blueprint; you'd end up with a mess, right? PEMDAS/BODMAS is our blueprint for math, ensuring we always arrive at the correct answer. This order prevents us from making common mistakes and keeps our calculations consistent and accurate. Think of it as the universal language of mathematics, ensuring everyone gets the same answer when solving the same problem.

Let’s break down what each letter stands for. Parentheses (or Brackets) come first. These are the round brackets ‘()’ or square brackets ‘[]’ that group parts of an expression together. We tackle everything inside these before anything else. Next up are Exponents (or Orders), which are those little numbers floating up high, indicating powers or roots. After exponents, we handle Multiplication and Division. Here’s a neat trick: we do these from left to right, as they appear in the expression. They're equal partners, so neither takes precedence over the other. Lastly, we have Addition and Subtraction, and just like multiplication and division, we perform these from left to right. So, with PEMDAS/BODMAS in our toolkit, we’re all set to dissect our expression. Remember, this isn't just a set of arbitrary rules; it's a logical system that ensures consistency and accuracy in mathematical calculations. By following PEMDAS/BODMAS, we avoid ambiguity and guarantee that everyone solving the same problem arrives at the same solution. Now, let's put this knowledge into action and see how it helps us solve our equation step by step!

Step 1: Simplifying Inside the Parentheses (2^3 + 6)

Alright, first things first, let’s zero in on what's happening inside the parentheses: (2^3 + 6). According to PEMDAS/BODMAS, we’ve got to handle anything inside parentheses before we even think about other operations. Inside our parentheses, we have two operations: an exponent (2^3) and addition (+ 6). Remembering our order of operations, exponents take precedence over addition. So, the very first thing we're going to tackle is 2^3. This little exponent tells us to multiply 2 by itself three times: 2 * 2 * 2. That's 2 multiplied by 2, which gives us 4, and then 4 multiplied by 2, which lands us at 8. So, 2^3 is equal to 8.

Now that we've conquered the exponent, our expression inside the parentheses looks like this: (8 + 6). This is much simpler! Now it's just a straightforward addition problem. Adding 8 and 6 is something we can easily do. 8 plus 6 equals 14. So, the expression inside the parentheses, (2^3 + 6), simplifies down to 14. We’ve just cleared the first hurdle! This step is a perfect example of how breaking down a complex problem into smaller, manageable parts makes it much less daunting. By focusing on one operation at a time, and following the order of operations, we ensure accuracy and avoid confusion. Now that we’ve simplified the parentheses, we’re ready to move on to the next step. We're making excellent progress, and the solution is getting closer with each step we complete. So, let's keep the momentum going and see what's next in our mathematical journey!

Step 2: Multiplication within the Brackets [2(14)]

Okay, guys, we've successfully simplified the expression inside the parentheses to 14. Our equation now looks like this: 1/2[2(14)] - 6. Next up, we need to deal with what’s happening inside the square brackets, which is [2(14)]. Remember, in math, when a number is right next to a set of parentheses or brackets, it usually means multiplication. So, 2(14) really means 2 multiplied by 14. This step is crucial because it follows directly from simplifying the parentheses, and it sets us up for the next operations in our sequence. Mastering this kind of step is vital for anyone looking to improve their math skills, as multiplication is a fundamental operation that appears in countless mathematical contexts.

Now, let’s do the multiplication. What is 2 multiplied by 14? You can think of it as 14 added to itself, or you can do it the traditional way. Either way, 2 times 14 equals 28. So, [2(14)] simplifies to 28. Our equation is getting simpler and simpler, isn't it? We’ve taken a seemingly complex part of the equation and reduced it to a single number. This is the power of following the order of operations – it helps us break down tough problems into manageable chunks. We're not just solving a problem here; we're building our mathematical muscles and gaining confidence in our ability to tackle challenges. With this step completed, we're one step closer to the final answer. Let's move on to the next part of our journey and see how we can further simplify our equation!

Step 3: Multiplying by 1/2: 1/2[28]

Fantastic work so far! We've simplified the expression inside the brackets, and now our equation reads 1/2[28] - 6. The next operation we need to tackle is multiplying by 1/2. Remember, 1/2[28] means 1/2 multiplied by 28. Multiplying by 1/2 is the same as dividing by 2, so we're essentially finding half of 28. This is a common type of calculation, and mastering it can be super handy in many situations, not just in math class but also in everyday life, like splitting a bill or measuring ingredients for a recipe.

So, what’s half of 28? If you think about it, 28 can be split into two 14s. Therefore, 1/2 multiplied by 28 equals 14. Our equation now looks like this: 14 - 6. We're almost there! We've managed to whittle down a somewhat complex expression into a very simple subtraction problem. This step highlights the beauty of the order of operations – by systematically addressing each part of the equation, we transform a potentially daunting task into a series of manageable steps. Each simplification brings us closer to the final answer, and it's incredibly satisfying to see the progress we're making. Now, let's finish strong and complete the final step to reveal the solution!

Step 4: Final Subtraction: 14 - 6

Alright, guys, we've reached the final stretch! Our equation has been beautifully simplified to 14 - 6. This is the last step, and it's a simple subtraction. We've navigated through parentheses, exponents, multiplication, and now we're here, ready to subtract. This final step is a testament to the power of methodical problem-solving. By following the order of operations, we've transformed a complex expression into a straightforward calculation. It's like climbing a mountain – each step brings you closer to the summit, and the view from the top is always worth the effort. In this case, the