Step-by-Step Guide Evaluate (-2(5+3)-9 ÷ 3+5) / (-3 × -5+-2 × 4)

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Hey guys! Let's break down this mathematical expression together and figure out the solution. We've got a fraction here, and like any good mathematical puzzle, we need to follow the order of operations to get it right. Remember PEMDAS/BODMAS? It's our trusty guide!

$\frac{-2(5+3)-9 \div 3+5}{-3 \times -5+-2 \times 4}$

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we jump into the nitty-gritty, let's quickly recap the order of operations. This is super important because it dictates the sequence in which we perform calculations.

  • Parentheses / Brackets: Anything inside parentheses or brackets comes first.
  • Exponents / Orders: Next up are exponents or orders (like squares and cubes).
  • Multiplication and Division: These are done from left to right.
  • Addition and Subtraction: These are also done from left to right.

Think of it like a recipe – you can't bake the cake before you mix the ingredients, right? The order of operations ensures we get the correct answer every time.

Step-by-Step Evaluation of the Numerator

Okay, let's tackle the numerator first. It looks a bit complex, but we'll break it down piece by piece. The numerator is: -2(5+3)-9 ÷ 3+5

1. Parentheses

The first thing we spot is the parentheses: (5+3). Let's handle that first.

5 + 3 = 8

So, now our numerator looks like this: -2(8) - 9 ÷ 3 + 5

2. Multiplication and Division (from left to right)

Next up, we've got multiplication and division. Remember, we do these from left to right.

First, let's do the multiplication: -2(8)

-2 * 8 = -16

Now our numerator is: -16 - 9 ÷ 3 + 5

Next, we handle the division: 9 ÷ 3

9 ÷ 3 = 3

So the numerator becomes: -16 - 3 + 5

3. Addition and Subtraction (from left to right)

Finally, we've got addition and subtraction. Again, we work from left to right.

First, let's do the subtraction: -16 - 3

-16 - 3 = -19

Now our numerator is: -19 + 5

Finally, the addition: -19 + 5

-19 + 5 = -14

So, the numerator simplifies to -14. Awesome! We're halfway there.

Step-by-Step Evaluation of the Denominator

Now, let's dive into the denominator. It's -3 × -5 + -2 × 4. Let's apply PEMDAS/BODMAS here as well.

1. Multiplication

We have two multiplication operations here, so we'll do them from left to right.

First, -3 × -5:

-3 * -5 = 15

Remember, a negative times a negative equals a positive! Now our denominator looks like this: 15 + -2 × 4

Next, we have -2 × 4:

-2 * 4 = -8

So the denominator becomes: 15 + -8

2. Addition

Now we just have one addition operation left:

15 + -8

This is the same as 15 - 8, so:

15 - 8 = 7

Therefore, the denominator simplifies to 7. We're on a roll!

Final Calculation

Okay, we've simplified both the numerator and the denominator. Now we can put them together and get our final answer. The original expression was:

$\frac{-2(5+3)-9 \div 3+5}{-3 \times -5+-2 \times 4}$

We found that the numerator is -14 and the denominator is 7. So, our fraction is:

$\frac{-14}{7}$

Now we just need to simplify this fraction. What number can divide both -14 and 7? You guessed it – 7!

-14 ÷ 7 = -2

7 ÷ 7 = 1

So, the simplified fraction is:

$\frac{-2}{1}$

And any number divided by 1 is just the number itself. So, our final answer is:

-2

Conclusion: The Final Answer

We did it! By carefully following the order of operations (PEMDAS/BODMAS), we successfully evaluated the expression. The final answer is -2. See, math can be fun when you break it down step by step!

Key takeaways:

  • Always remember PEMDAS/BODMAS.
  • Break down complex expressions into smaller, manageable steps.
  • Double-check your work, especially with negative numbers.

I hope this step-by-step guide was helpful! Keep practicing, and you'll become a math whiz in no time. Remember guys, practice makes perfect!

Understanding Mathematical Expressions

Mathematical expressions are the language of mathematics. They are combinations of numbers, variables, and mathematical operations that represent a value. Evaluating these expressions involves simplifying them to a single numerical value. This process often requires a strong understanding of the order of operations and the properties of mathematical operations.

Key Components of Mathematical Expressions:

  • Numbers: These are the basic building blocks, such as integers, fractions, decimals, and more.
  • Variables: These are symbols (usually letters) that represent unknown values.
  • Operators: These are symbols that indicate mathematical operations, such as addition (+), subtraction (-), multiplication (× or *), division (÷ or /), and exponentiation (^).
  • Parentheses/Brackets: These are used to group parts of an expression and indicate the order in which operations should be performed.

Importance of Understanding Mathematical Expressions:

Understanding mathematical expressions is crucial for various reasons:

  • Problem Solving: Many real-world problems can be modeled using mathematical expressions. Being able to evaluate and manipulate these expressions allows us to solve these problems.
  • Higher-Level Mathematics: Mathematical expressions form the foundation of more advanced topics like algebra, calculus, and statistics.
  • Critical Thinking: Evaluating expressions requires logical thinking and attention to detail, which are valuable skills in many areas of life.

Tips for Mastering Mathematical Expressions:

  • Practice Regularly: The more you practice, the more comfortable you'll become with evaluating expressions.
  • Review the Basics: Make sure you have a solid understanding of the order of operations and basic mathematical operations.
  • Break Down Complex Problems: Divide complex expressions into smaller, more manageable parts.
  • Check Your Work: Always double-check your calculations to avoid errors.
  • Seek Help When Needed: Don't be afraid to ask for help from teachers, tutors, or classmates if you're struggling.

By understanding the components of mathematical expressions and following these tips, you can improve your ability to evaluate them and succeed in mathematics.

Common Mistakes to Avoid When Evaluating Expressions

Evaluating mathematical expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  1. Ignoring the Order of Operations: This is the most frequent error. Always remember PEMDAS/BODMAS and perform operations in the correct order.
  2. Incorrectly Handling Negative Signs: Negative signs can be confusing, especially when combined with multiplication and division. Remember the rules: negative × negative = positive, negative × positive = negative, and so on.
  3. Misinterpreting Parentheses: Parentheses indicate the order of operations, so make sure you're evaluating the expressions inside them correctly.
  4. Arithmetic Errors: Simple addition, subtraction, multiplication, or division mistakes can throw off your entire calculation. Double-check your work!
  5. Forgetting to Distribute: When multiplying a number by an expression in parentheses, remember to distribute the number to each term inside the parentheses. For example, 2(x + 3) = 2x + 6.
  6. Combining Unlike Terms: You can only add or subtract like terms (terms with the same variable and exponent). For example, 2x + 3x can be combined to 5x, but 2x + 3 cannot.
  7. Incorrectly Applying Exponents: Make sure you understand how exponents work. For example, x^2 means x multiplied by itself (x * x).
  8. Not Simplifying Fractions: Always simplify your final answer to its lowest terms.
  9. Rushing Through the Problem: Take your time and work through each step carefully. Rushing can lead to careless errors.
  10. Not Checking Your Answer: If possible, check your answer by plugging it back into the original expression or using a different method to solve the problem.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in evaluating mathematical expressions.

Practice Problems to Sharpen Your Skills

Now that we've covered the basics and common mistakes, it's time to put your skills to the test! Here are a few practice problems to help you master evaluating expressions:

  1. Evaluate: 3(4 + 2) - 15 ÷ 3
  2. Simplify: -5(2 - 7) + 4 × -3
  3. Calculate: (10 - 4)2 ÷ 2 + 1
  4. Solve: -2(3 + 1) - 8 ÷ -4 + 5
  5. Find the value of: 12 ÷ (1 + 2) + 3 × (5 - 1)

Tips for Practice:

  • Work through each problem step by step, showing your work.
  • Use PEMDAS/BODMAS to guide your calculations.
  • Double-check your answers for accuracy.
  • If you get stuck, review the concepts and examples we discussed earlier.
  • Don't be afraid to ask for help if you need it.

Practice is key to mastering any mathematical skill. The more you work on evaluating expressions, the more comfortable and confident you'll become. So grab a pencil and paper, and let's get started! Remember, even if you make a mistake, it's an opportunity to learn and improve. Keep practicing, and you'll be a math pro in no time!