Solving Numerical Expressions: A Step-by-Step Guide

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Hey guys! Today, we're diving deep into the fascinating world of numerical expressions. We'll break down some problems step-by-step, making sure you grasp the core concepts. So, let's put on our math hats and get started!

a) Finding a Number 15 More Than 15 Squared

This problem involves understanding exponents and basic addition. Our main keyword here is numerical expressions, so let's see how it fits. We need to translate the word problem into a mathematical expression. The key phrase here is "15 more than 15 squared." First, let's figure out what "15 squared" means. Squaring a number means multiplying it by itself. So, 15 squared (written as 15²) is 15 * 15. Guys, can you calculate that? 15 * 15 equals 225. Now, the problem states the number we're looking for is 15 more than 15 squared. That means we need to add 15 to 225. Our numerical expression, therefore, looks like this: 15² + 15. Remember, numerical expressions are just combinations of numbers and mathematical operations! To solve it, we already know 15² is 225. So, we simply add 15: 225 + 15. And the answer is... 240! So, the number that is 15 more than 15 squared is 240. See? We took a word problem, translated it into a numerical expression, and solved it. It's all about breaking it down step by step. This kind of problem is really common, guys, so make sure you're comfortable with squares and addition. Now, let's move on to the next one!

b) Calculating a Number Less Than 9 Cubed and Doubling It

This problem takes things up a notch by introducing the concept of cubing a number and then doubling the result. Again, the key is to break it down into manageable steps. We're still dealing with numerical expressions, but now we have a couple more operations to consider. The first part of the problem states that we have a natural number that is 665 less than 9 cubed. Okay, let's tackle "9 cubed" first. Cubing a number means multiplying it by itself three times. So, 9 cubed (written as 9³) is 9 * 9 * 9. What's that, guys? 9 * 9 is 81, and 81 * 9 is 729. So, 9³ equals 729. Now, the problem says our number is 665 less than 9 cubed. This means we need to subtract 665 from 729. Our numerical expression for this part is: 9³ - 665. Let's calculate that: 729 - 665 equals 64. But we're not done yet! The problem asks for double the value of this number. Doubling a number means multiplying it by 2. So, we need to multiply 64 by 2. Our complete numerical expression is: (9³ - 665) * 2. We already know 9³ - 665 is 64, so we just need to do 64 * 2. And that equals 128! Therefore, double the value of the number that is 665 less than 9 cubed is 128. Guys, you see how we built this up? We dealt with the cubing first, then the subtraction, and finally the multiplication. That's the key to tackling complex numerical expressions.

c) Evaluating a Multi-Operation Numerical Expression

Alright, guys, this one looks a bit intimidating, but don't worry! We'll conquer it using the order of operations – PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This problem really highlights the importance of understanding numerical expressions and how the order of operations dictates how we solve them. Our expression is: $((300 : 100 + 5^2) : 4) * 10 - 100. Let's break it down step-by-step:

  1. Parentheses: We have nested parentheses, so we start with the innermost one: 300 : 100 + 5². Within this, we have division and an exponent. According to PEMDAS, we do the exponent first: 5² is 5 * 5, which equals 25. Now our innermost parenthesis looks like this: 300 : 100 + 25.
  2. Division: Next up is division: 300 : 100 equals 3. So, our innermost parenthesis now is: 3 + 25.
  3. Addition: We complete the innermost parenthesis by adding: 3 + 25 equals 28. Now our main expression is: (28:4)∗10−100(28 : 4) * 10 - 100.
  4. Parentheses (again): We have another set of parentheses to deal with: 28 : 4 equals 7. Our expression is now: 7 * 10 - 100.
  5. Multiplication: Next, we multiply: 7 * 10 equals 70. Our expression simplifies to: 70 - 100.
  6. Subtraction: Finally, we subtract: 70 - 100 equals -30.

So, the final answer to the expression ((300:100+52):4)∗10−100((300 : 100 + 5^2) : 4) * 10 - 100 is -30! Guys, that was a journey, but we made it! We carefully followed PEMDAS, tackled each operation in the right order, and arrived at the solution. This problem really showcases how important it is to pay attention to the order of operations when dealing with numerical expressions.

Why Understanding Numerical Expressions Matters

Understanding numerical expressions is crucial for so many reasons! It's not just about solving math problems in school, guys. These skills are fundamental for:

  • Everyday Life: From calculating discounts at the store to figuring out how much paint you need for a room, numerical expressions pop up everywhere. Think about splitting a bill with friends – you're using numerical expressions to figure out each person's share!
  • Higher Math: Algebra, calculus, and even more advanced math topics build upon the foundation of numerical expressions. If you're solid on the basics, you'll have a much easier time tackling more complex concepts.
  • Problem-Solving: The ability to translate real-world scenarios into numerical expressions and solve them is a powerful problem-solving skill that applies to many areas of life.
  • Critical Thinking: Working with numerical expressions helps you develop critical thinking skills, like analyzing information, identifying relevant details, and breaking down complex problems into smaller steps. Guys, these are skills that will serve you well in any field!

Tips for Mastering Numerical Expressions

Here are a few tips to help you become a pro at solving numerical expressions:

  • Practice, practice, practice! The more you work with numerical expressions, the more comfortable you'll become. Start with simpler problems and gradually work your way up to more complex ones.
  • Master the order of operations (PEMDAS/BODMAS). This is absolutely crucial! Always follow the correct order to avoid errors.
  • Break down complex problems. Don't try to do everything at once. Break the expression down into smaller, more manageable steps.
  • Show your work. Writing down each step helps you stay organized and makes it easier to identify any mistakes.
  • Check your answers. If possible, use a calculator or other method to check your work and ensure you've arrived at the correct solution.

Guys, solving numerical expressions might seem daunting at first, but with practice and a systematic approach, you can absolutely master them. Remember to break problems down, follow the order of operations, and most importantly, don't be afraid to ask for help when you need it. Keep practicing, and you'll be solving complex numerical expressions like a pro in no time!