Calculate 30 + [6² ÷ (5 - 3) + 1] A Step-by-Step Solution

Hey guys! Let's break down this mathematical expression step-by-step to make sure we get the right answer. Don't worry, it might look a little intimidating at first, but we'll tackle it together using the order of operations. This is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into the calculation, it's super important to understand the order of operations. It's like the golden rule of math – we have to follow it to get the correct result. PEMDAS (or BODMAS, which is the same thing but uses a different acronym: Brackets, Orders, Division and Multiplication, Addition and Subtraction) tells us the sequence in which we should perform mathematical operations:

1. Parentheses (or Brackets): First, we deal with anything inside parentheses or brackets.
2. Exponents (or Orders): Next up are exponents, like squares and cubes.
3. Multiplication and Division: These operations are performed from left to right.
4. Addition and Subtraction: Finally, we do addition and subtraction, also from left to right.

Following this order ensures that everyone arrives at the same answer, no matter who's doing the math. Think of it as a universal language for numbers!

Why is Order of Operations Important?

You might be wondering, “Why can’t I just do the operations in any order?” Well, imagine if we didn’t have this rule. Someone might try to add before multiplying, leading to a completely different and incorrect answer. The order of operations provides a consistent and logical way to solve mathematical problems, preventing ambiguity and ensuring accuracy. It’s the foundation upon which more complex math is built, so mastering it is crucial for success in mathematics.

For example, let’s say we have the expression 2 + 3 * 4. If we simply go from left to right, we might do 2 + 3 = 5 first, and then multiply by 4, giving us 20. But, if we follow PEMDAS, we need to do the multiplication first: 3 * 4 = 12. Then, we add 2: 2 + 12 = 14. See the difference? The correct answer is 14, highlighting the importance of adhering to the order of operations.

In the expression we’re tackling today, 30 + [6² ÷ (5 - 3) + 1], we’ll see how PEMDAS comes into play at each step, guiding us towards the correct solution. So, keep PEMDAS in mind as we proceed, and you’ll find that even complex-looking expressions become manageable.

Step 1: Tackling the Parentheses (5 - 3)

The first thing we need to do, according to PEMDAS, is to deal with the expression inside the parentheses: (5 - 3). This is a straightforward subtraction, so let's get it done!

5 - 3 = 2

So, the expression inside the parentheses simplifies to 2. Now, we can rewrite our original expression, replacing (5 - 3) with 2:

30 + [6² ÷ 2 + 1]

We've taken the first step in simplifying the expression, and it already looks a bit less intimidating, doesn't it? By focusing on the parentheses first, we've made the rest of the calculation much easier to manage. This is the power of following the order of operations – it breaks down complex problems into smaller, more manageable steps.

Why Parentheses Come First

Parentheses act like a VIP section in the order of operations. They tell us, “Hey, do this part first!” This is crucial because operations within parentheses often change the outcome of the entire expression. They group terms together, indicating that those terms should be treated as a single entity before any other calculations are performed.

In our example, if we didn't calculate (5 - 3) first, we'd be diving into exponents and division before we were ready, potentially leading to a wrong answer. By prioritizing parentheses, we ensure that we’re working with the correct values at each stage of the calculation. It’s like building a house – you need to lay the foundation (the parentheses) before you start putting up the walls (the other operations).

So, remember, whenever you see parentheses in a mathematical expression, that's your cue to start there. It's the first step in unraveling the problem and setting yourself up for success.

Step 2: Evaluating the Exponent (6²)

Now that we've handled the parentheses, let's move on to the next operation in PEMDAS: exponents. In our expression, 30 + [6² ÷ 2 + 1], we have one exponent to deal with: 6². Remember, 6² means 6 multiplied by itself, or 6 * 6.

6² = 6 * 6 = 36

So, 6² equals 36. We can now substitute 36 for 6² in our expression:

30 + [36 ÷ 2 + 1]

We're making great progress! By evaluating the exponent, we've further simplified the expression, bringing us closer to the final answer. It's like peeling away the layers of an onion – each step gets us closer to the core.

The Power of Exponents

Exponents are a shorthand way of representing repeated multiplication. Instead of writing 6 * 6, we can simply write 6². This not only saves space but also makes it easier to understand and work with larger numbers. Exponents are used extensively in various fields, from science and engineering to finance and computer science.

Understanding exponents is crucial for grasping more advanced mathematical concepts, such as exponential growth and decay, scientific notation, and logarithms. They play a fundamental role in describing phenomena that change rapidly over time, like population growth or compound interest.

In our current problem, evaluating the exponent 6² was a critical step in simplifying the expression. By correctly calculating 6² as 36, we ensured that we were working with the right values in the subsequent operations. This highlights the importance of mastering exponents and their role in the order of operations.

Step 3: Performing the Division (36 ÷ 2)

With the parentheses and exponents out of the way, we now turn our attention to multiplication and division, working from left to right. In our expression, 30 + [36 ÷ 2 + 1], we have one division operation: 36 ÷ 2. Let's perform this calculation.

36 ÷ 2 = 18

So, 36 divided by 2 equals 18. We can now replace 36 ÷ 2 with 18 in our expression:

30 + [18 + 1]

We're getting closer and closer to the final answer! By performing the division, we've continued to simplify the expression, making the remaining steps much easier to manage. This is a testament to the power of breaking down complex problems into smaller, more manageable parts.

Why Division and Multiplication Have Equal Priority

You might notice that division and multiplication are on the same level in PEMDAS. This means that we perform these operations from left to right, as they appear in the expression. There's no inherent rule that says division should always come before multiplication, or vice versa. It's all about the order in which they show up.

The same principle applies to addition and subtraction, which we'll see in the next step. This left-to-right approach ensures that we're consistently applying the order of operations and avoiding any ambiguity in our calculations.

In our example, if we had both multiplication and division, we would simply work our way from left to right, performing each operation as we encounter it. This simple rule helps keep things clear and prevents errors in our calculations.

Step 4: Completing the Addition Inside the Brackets (18 + 1)

Now we're cooking! We've handled the parentheses, exponents, and division. Next up, we need to tackle the addition inside the brackets: [18 + 1]. This is a simple addition, so let's do it.

18 + 1 = 19

So, 18 plus 1 equals 19. We can now replace [18 + 1] with 19 in our expression:

30 + 19

Look how far we've come! The expression has been simplified to just a single addition. We're almost at the finish line!

Brackets as Grouping Symbols

Just like parentheses, brackets act as grouping symbols, telling us to perform the operations inside them before anything else outside. In this case, the brackets ensured that we added 18 and 1 before adding the result to 30. Grouping symbols are essential for controlling the order of operations and ensuring that we get the correct answer.

There are different types of grouping symbols, including parentheses (), brackets [], and braces {}. They all serve the same purpose – to indicate which operations should be performed first. When you have nested grouping symbols (one inside another), you typically work from the innermost to the outermost.

In our expression, the brackets grouped the addition of 18 and 1, ensuring that this operation was performed before the final addition with 30. This is a crucial step in following the order of operations and arriving at the correct solution.

Step 5: The Final Addition (30 + 19)

Alright, guys, we've made it to the final step! We've simplified the expression all the way down to a simple addition: 30 + 19. Let's perform this final calculation to get our answer.

30 + 19 = 49

So, 30 plus 19 equals 49. We did it! The answer to the expression 30 + [6² ÷ (5 - 3) + 1] is 49.

The Importance of Reviewing Your Work

In mathematics, as in many areas of life, it's always a good idea to double-check your work. After reaching the final answer, take a moment to review each step of the calculation to ensure that you haven't made any mistakes. This is especially important in more complex problems, where errors can easily creep in.

You can review your work by going back through each step and verifying that you've performed the operations correctly. You can also try plugging the answer back into the original expression to see if it makes sense. If everything checks out, you can be confident that you've arrived at the correct solution.

In our example, we can quickly review the steps we took: parentheses, exponents, division, addition inside brackets, and the final addition. By confirming that each step was performed correctly, we can be sure that our answer of 49 is accurate.

Conclusion: Mastering the Order of Operations

So, there you have it! We've successfully calculated the expression 30 + [6² ÷ (5 - 3) + 1] and arrived at the answer: 49. We did this by carefully following the order of operations (PEMDAS/BODMAS), breaking the problem down into manageable steps, and double-checking our work along the way.

Mastering the order of operations is crucial for success in mathematics. It's the foundation upon which more complex concepts are built. By understanding and applying PEMDAS, you can confidently tackle a wide range of mathematical problems, from simple arithmetic to more advanced algebra and calculus.

Keep Practicing!

The best way to solidify your understanding of the order of operations is to practice. Work through different examples, starting with simpler expressions and gradually moving on to more complex ones. The more you practice, the more comfortable and confident you'll become.

Remember, math is like any other skill – it takes time and effort to develop. Don't get discouraged if you encounter challenges along the way. Embrace the process, ask questions, and keep practicing. With persistence and a solid understanding of the fundamentals, you can achieve your mathematical goals. You got this!