Solving 12⅖-7×14+210÷³√8 A Step-by-Step Guide

Hey guys! Ever stumbled upon a mathematical expression that looks like a jumbled mess of numbers and operations? Well, you're not alone! Math problems can sometimes seem daunting, but fear not! We're about to break down a seemingly complex equation into simple, manageable steps. Today, we're tackling the expression 12⅖ - 7 × 14 + 210 ÷ ³√8. This might look intimidating at first glance, but with a bit of order of operations and some mathematical finesse, we'll conquer it together.

Demystifying the Order of Operations

Before we dive into the specifics of our problem, it's crucial to understand the golden rule of mathematical expressions: the order of operations. This is the secret sauce that ensures we all arrive at the same correct answer. Think of it as the mathematical traffic laws – we need to follow them to avoid any chaotic collisions! The most common mnemonic device to remember this order is PEMDAS, which stands for:

• Parentheses (and other grouping symbols)
• Exponents (and roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This order dictates the sequence in which we perform the operations. We first tackle anything inside parentheses or grouping symbols, then exponents and roots, followed by multiplication and division (working from left to right), and finally, addition and subtraction (again, from left to right). Mastering PEMDAS is like having a superpower in the world of math – it allows you to dissect complex expressions with confidence and accuracy.

In our particular expression, 12⅖ - 7 × 14 + 210 ÷ ³√8, we have a mix of operations. We have subtraction, multiplication, addition, division, and a cube root. According to PEMDAS, we'll need to address the cube root first, then handle the multiplication and division, and finally, wrap things up with addition and subtraction. Let's begin!

Step 1: Conquering the Cube Root

The first task on our PEMDAS checklist is to simplify the cube root: ³√8. This might sound fancy, but it's simply asking: "What number, when multiplied by itself three times, equals 8?" If you're familiar with your cubes, you might already know the answer. If not, let's think it through. We know that 2 × 2 × 2 = 8. Therefore, the cube root of 8 is 2. We can confidently replace ³√8 with 2 in our expression.

Now, our expression looks a little less intimidating: 12⅖ - 7 × 14 + 210 ÷ 2. We've successfully tackled the cube root, and we're one step closer to solving the entire problem. This step highlights the importance of understanding roots and exponents. They often form the foundation for more complex calculations, and mastering them makes the journey much smoother.

Step 2: Multiplication and Division – Left to Right

With the cube root handled, PEMDAS guides us to the next operations: multiplication and division. Remember, we perform these from left to right. This is a crucial point – we can't just pick and choose which one to do first. We need to follow the order meticulously. In our expression, 12⅖ - 7 × 14 + 210 ÷ 2, we encounter multiplication first: 7 × 14. Let's calculate this.

7 multiplied by 14 equals 98. So, we can replace 7 × 14 with 98 in our expression. Now we have: 12⅖ - 98 + 210 ÷ 2. Next up is the division: 210 ÷ 2. This is a straightforward division problem. 210 divided by 2 equals 105. Let's substitute this into our expression: 12⅖ - 98 + 105. We've successfully navigated the multiplication and division operations, and our expression is becoming even simpler.

Step 3: Addition and Subtraction – From Left to Right

Now we're in the home stretch! PEMDAS tells us to perform addition and subtraction, again working from left to right. Our expression currently reads: 12⅖ - 98 + 105. Before we jump into the arithmetic, let's convert the mixed number 12⅖ into an improper fraction. This will make the subtraction and addition steps easier to manage.

To convert 12⅖ to an improper fraction, we multiply the whole number (12) by the denominator (5) and add the numerator (2). This gives us (12 × 5) + 2 = 60 + 2 = 62. We then place this result over the original denominator, giving us 62/5. Now our expression looks like this: 62/5 - 98 + 105. To perform the subtraction and addition, we need to find a common denominator. Since we have a fraction with a denominator of 5 and whole numbers, we can express the whole numbers as fractions with a denominator of 5 as well.

98 can be written as 98/1, and to get a denominator of 5, we multiply both numerator and denominator by 5: (98 × 5) / (1 × 5) = 490/5. Similarly, 105 can be written as 105/1, and to get a denominator of 5, we multiply both numerator and denominator by 5: (105 × 5) / (1 × 5) = 525/5. Now our expression is: 62/5 - 490/5 + 525/5. We can now perform the subtraction and addition: 62/5 - 490/5 = -428/5. Then, -428/5 + 525/5 = 97/5. So, the result is 97/5. To convert this improper fraction back to a mixed number, we divide 97 by 5. 97 divided by 5 is 19 with a remainder of 2. Therefore, 97/5 is equal to 19⅖.

Final Answer: The Solution to the Puzzle

After carefully following the order of operations and performing each step with precision, we've arrived at our final answer: 19⅖. This journey highlights the power of breaking down complex problems into smaller, more manageable steps. By understanding PEMDAS and applying it consistently, we can tackle even the most intimidating mathematical expressions with confidence.

So, guys, the next time you encounter a mathematical expression that looks like a tangled mess, remember the steps we've covered today. Embrace the order of operations, take it one step at a time, and you'll be surprised at how easily you can unlock the solution. Math might seem like a puzzle at times, but with the right approach, it's a puzzle we can all solve!

Key Takeaways

• PEMDAS is your best friend: Always remember the order of operations (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). It's the roadmap to solving any mathematical expression.
• Break it down: Complex problems become much simpler when you break them down into smaller, manageable steps.
• Practice makes perfect: The more you practice, the more comfortable you'll become with mathematical operations and the order in which to perform them.
• Don't be afraid to ask for help: If you're stuck, don't hesitate to seek assistance from teachers, classmates, or online resources. Learning is a collaborative process!

Remember, math is a journey, not a destination. Enjoy the process of learning, and celebrate your successes along the way. You've got this!

Real-World Applications of Order of Operations

The order of operations isn't just some abstract concept confined to math textbooks. It's a fundamental principle that underpins many real-world calculations and problem-solving scenarios. From calculating the total cost of a shopping trip to programming a complex algorithm, the order in which we perform operations matters.

For example, consider a simple budgeting scenario. Let's say you have a weekly income, and you want to calculate how much money you have left after paying for your expenses. You might need to subtract your rent, utility bills, and grocery expenses from your income. The order in which you perform these subtractions will determine the final amount you have left. If you accidentally add your expenses instead of subtracting them, you'll end up with a very inaccurate picture of your financial situation!

In the world of computer programming, the order of operations is even more critical. Programming languages rely on strict rules of precedence to interpret mathematical expressions. If you write code that doesn't follow the correct order of operations, your program might produce unexpected results or even crash. This is why programmers need to have a solid understanding of mathematical principles, including PEMDAS.

Even in everyday situations like cooking, the order of operations can play a role. If a recipe calls for you to add ingredients in a specific sequence, following that order is essential for the dish to turn out correctly. For instance, if you add the liquid ingredients before the dry ingredients, you might end up with a lumpy batter. The order in which you perform these actions matters, just like in mathematics.

Common Mistakes to Avoid

While the order of operations might seem straightforward, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Forgetting PEMDAS: This is the most common mistake of all. Always keep PEMDAS in mind when tackling mathematical expressions. It's helpful to write it down or have it readily accessible as a reminder.
• Incorrectly handling subtraction: Subtraction can be tricky, especially when dealing with negative numbers. Remember that subtracting a negative number is the same as adding its positive counterpart.
• Ignoring parentheses: Parentheses are like VIPs in the order of operations. They demand your attention first. Make sure to simplify everything inside the parentheses before moving on to other operations.
• Misinterpreting exponents and roots: Exponents and roots are powerful operations, but they can be confusing if you don't understand their meaning. Remember that an exponent indicates how many times to multiply a number by itself, while a root is the inverse operation of an exponent.
• Not working from left to right: When performing multiplication and division (or addition and subtraction), remember to work from left to right. This is crucial for arriving at the correct answer.

By being aware of these common mistakes, you can significantly reduce the likelihood of errors in your calculations. Always double-check your work and pay close attention to each step in the process.