Solve: $15 \div 3+(6-1)^2$ - Step-by-Step Guide

by Sebastian Müller 48 views

Hey guys! Math can sometimes seem like a puzzle, but don't worry, we're here to break it down step by step. Today, we're going to tackle an expression that might look a little intimidating at first glance, but I promise it's totally manageable. Our mission is to find the value of this expression: 15÷3+(6−1)215 \div 3+(6-1)^2. Sounds like fun, right? Let's dive in!

Understanding the Order of Operations

Before we even think about plugging in numbers and doing calculations, we need to talk about the golden rule of math expressions: the order of operations. This is basically a set of instructions that tells us which parts of the expression to solve first. Think of it as a mathematical GPS – it makes sure we get to the right answer every time. The most common way to remember the order of operations is the acronym PEMDAS, which stands for:

  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

So, when we look at our expression, 15÷3+(6−1)215 \div 3+(6-1)^2, PEMDAS tells us exactly where to start. First up? You guessed it, the parentheses!

Step 1: Tackling the Parentheses

Parentheses are like VIP sections in the math world – they get priority treatment. Inside our parentheses, we have (6-1). This is a straightforward subtraction problem. Six minus one is five. So, we can simplify our expression to:

15÷3+(5)215 \div 3 + (5)^2

See? We're already making progress! The expression is looking a bit less scary now. We've handled the parentheses, so what's next on our PEMDAS list? That's right, it's exponents!

Step 2: Unleashing the Power of Exponents

Exponents are a way of showing repeated multiplication. When we see something like (5)2(5)^2, it means we need to multiply 5 by itself. In other words, (5)2(5)^2 is the same as 5∗55 * 5. And what's 5∗55 * 5? It's 25! So, we can replace (5)2(5)^2 with 25 in our expression:

15÷3+2515 \div 3 + 25

Awesome! We've conquered the exponents. Now, we're left with division and addition. Remember what PEMDAS tells us about these operations? We need to do them from left to right. So, which comes first?

Step 3: Diving into Division

Looking at our expression, 15÷3+2515 \div 3 + 25, we see that division comes before addition when reading from left to right. So, let's handle the division first. Fifteen divided by three is five. We can replace 15÷315 \div 3 with 5, giving us:

5+255 + 25

We're almost there, guys! Just one more step to go. We've simplified the expression down to a simple addition problem. Let's wrap it up!

Step 4: Adding It All Up

Our final step is to add 5 and 25 together. What do we get? Thirty! That's it! We've found the value of the expression.

The Final Answer

So, after following the order of operations and carefully working through each step, we've discovered that the value of the expression 15÷3+(6−1)215 \div 3+(6-1)^2 is 30.

Congratulations! You did it! Math expressions might seem tricky at first, but with a little practice and a solid understanding of PEMDAS, you can solve anything. Remember to take it one step at a time, and don't be afraid to break things down. Keep up the great work, and I'll see you in the next math adventure!

Why Understanding the Order of Operations Matters

Alright, so we successfully navigated through the expression 15÷3+(6−1)215 \div 3 + (6-1)^2 and arrived at the correct answer of 30. But you might be wondering, “Why is this order of operations thing so important anyway?” Well, imagine if we didn't have a set of rules to follow. What if one person decided to add before dividing, and another person decided to do the exponent last? We'd end up with a whole bunch of different answers for the same problem! That's where the order of operations comes in – it ensures that everyone solves the expression in the same way, so we all get the same, correct answer.

Think of it like following a recipe. If you add the ingredients in the wrong order, you might end up with a cake that's a total flop. The order of operations is like a recipe for math – it guarantees that your mathematical “cake” will turn out perfectly every time. Without it, math would be a chaotic mess!

Real-World Applications

The order of operations isn't just some abstract concept we learn in math class. It's actually used all the time in the real world, especially in fields like:

  • Computer Programming: Computers follow very strict rules, and the order of operations is crucial for writing code that works correctly. If you want a computer to perform a calculation, you need to tell it exactly what to do in the right order.
  • Engineering: Engineers use mathematical expressions all the time to design bridges, buildings, and machines. They need to be absolutely sure their calculations are accurate, and the order of operations helps them do that.
  • Finance: Calculating interest, taxes, and investments all involve mathematical expressions that require a specific order of operations. If you want to manage your money wisely, it's important to understand how these calculations work.
  • Everyday Life: Even in everyday situations, we use the order of operations without even realizing it. For example, if you're calculating the total cost of buying several items at a store, you're implicitly using the order of operations.

So, learning about PEMDAS isn't just about getting good grades in math class. It's about developing a fundamental skill that will be useful in many different areas of your life. The better you understand the order of operations, the more confident and successful you'll be at solving mathematical problems – both inside and outside the classroom.

Common Mistakes to Avoid

Now that we've walked through the steps and understand why the order of operations is so important, let's talk about some common mistakes people make when solving expressions. Knowing these pitfalls can help you avoid them and boost your math skills even further.

Forgetting PEMDAS

The most common mistake is simply forgetting the order of operations. It's easy to get caught up in the numbers and operations and just start solving from left to right, but that will often lead to the wrong answer. Always remind yourself of PEMDAS before you start, and double-check that you're following the correct order as you go.

Mixing Up Multiplication and Division (or Addition and Subtraction)

Remember that multiplication and division have the same priority, and we solve them from left to right. Similarly, addition and subtraction have the same priority and are also solved from left to right. A common mistake is to always do multiplication before division, or addition before subtraction, regardless of their order in the expression. Pay close attention to the order in which these operations appear and solve them accordingly.

Misunderstanding Exponents

Exponents can sometimes be confusing. Remember that an exponent tells you how many times to multiply a number by itself. So, 525^2 means 5∗55 * 5, not 5∗25 * 2. Make sure you understand the meaning of exponents and how to calculate them correctly.

Neglecting Parentheses

Parentheses are like little mathematical force fields – they tell you exactly what to solve first. Don't ignore them or try to solve other parts of the expression before dealing with the parentheses. Anything inside parentheses should be your top priority.

Careless Calculation Errors

Even if you understand the order of operations perfectly, a simple calculation error can throw off your entire answer. Double-check your work as you go, and be careful with your arithmetic. It's easy to make a mistake, especially when dealing with larger numbers or more complex expressions.

Not Showing Your Work

This might seem like a small thing, but showing your work is incredibly important. It helps you keep track of what you've done, and it makes it easier to spot any mistakes you might have made. Plus, if you do make a mistake, showing your work can help you (or your teacher) figure out where you went wrong.

By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering the order of operations and solving math expressions with confidence. Keep practicing, and don't be afraid to ask for help when you need it. You've got this!

Practice Makes Perfect: More Expressions to Try

Okay, guys, we've covered a lot of ground. We've learned about the order of operations (PEMDAS), walked through a step-by-step example, discussed why it's important, and even talked about common mistakes to avoid. Now, it's time to put your knowledge to the test! The best way to truly master the order of operations is to practice, practice, practice. So, let's try a few more expressions together. Grab a pencil and paper, and let's dive in!

Expression 1: 2∗(8−3)+42÷22 * (8 - 3) + 4^2 \div 2

Let's break this down using PEMDAS:

  1. Parentheses: First, we solve what's inside the parentheses: (8 - 3) = 5. So, our expression becomes: 2 * 5 + 4^2 \div 2
  2. Exponents: Next up is the exponent: 4^2 = 4 * 4 = 16. Our expression now looks like this: 2 * 5 + 16 \div 2
  3. Multiplication and Division (from left to right): We have both multiplication and division, so we work from left to right. First, 2 * 5 = 10. Then, 16 \div 2 = 8. Our expression is now: 10 + 8
  4. Addition: Finally, we add: 10 + 8 = 18

So, the value of the expression 2∗(8−3)+42÷22 * (8 - 3) + 4^2 \div 2 is 18.

Expression 2: 33−10÷5+13^3 - 10 \div 5 + 1

Let's tackle this one:

  1. Exponents: We start with the exponent: 3^3 = 3 * 3 * 3 = 27. The expression becomes: 27 - 10 \div 5 + 1
  2. Division: Next, we do the division: 10 \div 5 = 2. The expression is now: 27 - 2 + 1
  3. Subtraction and Addition (from left to right): We have both subtraction and addition, so we work from left to right. First, 27 - 2 = 25. Then, 25 + 1 = 26

So, the value of the expression 33−10÷5+13^3 - 10 \div 5 + 1 is 26.

Expression 3: 100÷(4+6)∗2−5100 \div (4 + 6) * 2 - 5

Let's try one more:

  1. Parentheses: We start with the parentheses: (4 + 6) = 10. The expression becomes: 100 \div 10 * 2 - 5
  2. Division and Multiplication (from left to right): We have both division and multiplication, so we work from left to right. First, 100 \div 10 = 10. Then, 10 * 2 = 20. The expression is now: 20 - 5
  3. Subtraction: Finally, we subtract: 20 - 5 = 15

So, the value of the expression 100÷(4+6)∗2−5100 \div (4 + 6) * 2 - 5 is 15.

How did you do, guys? Hopefully, you're starting to feel more comfortable with the order of operations. Remember, the key is to take it one step at a time, follow PEMDAS, and double-check your work. The more you practice, the easier it will become. Keep up the awesome work, and you'll be a math whiz in no time!

If you want even more practice, try making up your own expressions and solving them. You can also find tons of practice problems online or in math textbooks. Don't be afraid to challenge yourself – the more you push yourself, the more you'll learn. And most importantly, have fun with it! Math can be a fascinating and rewarding subject, and I'm so glad you're on this journey with me.