Solving Math Expressions: Step-by-Step Guide

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Solving Math Expressions: Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a cool math problem: −35×710+(−25+12)2-\frac{3}{5} \times \frac{7}{10}+\left(-\frac{2}{5}+\frac{1}{2}\right)^2. Don't worry, it looks a bit intimidating at first glance, but trust me, we'll break it down step-by-step to make it super easy to understand. This guide is all about evaluating expressions, a fundamental skill in mathematics. Whether you're a student, a math lover, or just someone who wants to brush up on their skills, this is for you. We'll go through the order of operations, simplify fractions, and make sure you're comfortable with both positive and negative numbers. By the end of this, you will have a solid grasp of how to approach and solve this type of math problem. So, let's get started and make math fun!

Understanding the Order of Operations: The Foundation

Before we start with the math problem, it's essential to understand the order of operations, often remembered by the acronym PEMDAS (or sometimes BODMAS). PEMDAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order is super important because it tells us the sequence in which we need to perform the calculations. If we do things in the wrong order, we will get the wrong answer! The key to successful problem-solving in math is knowing where to start.

So, let's break it down further. First, we need to deal with anything inside parentheses or brackets. This is always the first step. Next, we handle any exponents or powers. Then we move on to multiplication and division, working from left to right. Finally, we tackle addition and subtraction, also from left to right. Remembering PEMDAS is like having a roadmap for your calculations, ensuring you get the correct answer every time. Think of it like this: Parentheses are like the VIP section – you always deal with them first! Exponents are the next level, followed by the more general operations of multiplication, division, addition, and subtraction. Each step builds on the previous one, so paying close attention to the order is very important. Understanding and applying the order of operations is one of the most critical aspects of solving complex math problems and prevents common errors. Practice and familiarity with this order will make you more confident in solving a variety of expressions. Keep this in mind, and you're well on your way to mastering these kinds of problems.

Step-by-Step Solution: Breaking Down the Expression

Alright, guys, let's dive into solving −35×710+(−25+12)2-\frac{3}{5} \times \frac{7}{10}+\left(-\frac{2}{5}+\frac{1}{2}\right)^2 step-by-step. This is where the fun begins! We'll follow PEMDAS to keep us on track. Remember, the goal is to break down the problem into smaller, manageable chunks. This approach not only helps us find the answer but also helps us understand the underlying principles.

Step 1: Solving Parentheses

First, we need to deal with what's inside the parentheses: (−25+12)\left(-\frac{2}{5}+\frac{1}{2}\right). To add these fractions, we need a common denominator. The least common denominator (LCD) for 5 and 2 is 10. So, we'll convert both fractions to have a denominator of 10:

  • −25-\frac{2}{5} becomes −410-\frac{4}{10} (multiply the numerator and denominator by 2)
  • 12\frac{1}{2} becomes 510\frac{5}{10} (multiply the numerator and denominator by 5)

Now, add the fractions:

  • −410+510=110-\frac{4}{10} + \frac{5}{10} = \frac{1}{10}

So, the expression inside the parentheses simplifies to 110\frac{1}{10}. This first step is essential because it simplifies the problem. By focusing on the parentheses first, we reduce the complexity and get closer to our solution. Taking it one step at a time helps minimize errors and allows us to focus on each individual part of the problem. Remember, always start with what's inside the parentheses! This initial simplification sets the stage for the rest of the calculation.

Step 2: Evaluating the Exponent

Next, we need to evaluate the exponent. Our expression now looks like this: −35×710+(110)2-\frac{3}{5} \times \frac{7}{10} + (\frac{1}{10})^2. The term (110)2(\frac{1}{10})^2 means 110×110\frac{1}{10} \times \frac{1}{10}.

  • 110×110=1100\frac{1}{10} \times \frac{1}{10} = \frac{1}{100}

So, (110)2=1100(\frac{1}{10})^2 = \frac{1}{100}. Now, the expression becomes −35×710+1100-\frac{3}{5} \times \frac{7}{10} + \frac{1}{100}. This step is pretty straightforward, but it's important to remember what an exponent means. It means multiplying the number by itself as many times as the exponent indicates. Keep this in mind, and it'll be a piece of cake.

Step 3: Performing Multiplication

Now, let's take care of the multiplication. We have −35×710-\frac{3}{5} \times \frac{7}{10}. Multiply the numerators and the denominators separately:

  • −35×710=−3×75×10=−2150-\frac{3}{5} \times \frac{7}{10} = -\frac{3 \times 7}{5 \times 10} = -\frac{21}{50}

Our expression now is −2150+1100- \frac{21}{50} + \frac{1}{100}. Remember to multiply the numerators and denominators to perform multiplication. Also, note that multiplying a negative number by a positive number results in a negative number. Always watch your signs; they can make or break your answer.

Step 4: Adding the Fractions

Finally, we add the two fractions, −2150+1100- \frac{21}{50} + \frac{1}{100}. To add these fractions, we need a common denominator. The least common denominator for 50 and 100 is 100.

  • Convert −2150- \frac{21}{50} to a fraction with a denominator of 100 by multiplying the numerator and denominator by 2: −21×250×2=−42100- \frac{21 \times 2}{50 \times 2} = -\frac{42}{100}

Now, add the fractions:

  • −42100+1100=−41100- \frac{42}{100} + \frac{1}{100} = -\frac{41}{100}

So, the final answer is −41100- \frac{41}{100}. Congratulations, you've solved the expression! The final step of addition requires ensuring that you are working with a common denominator. By adding the fractions with the same denominator, you are able to arrive at the correct answer.

Conclusion: Mastering Expression Evaluation

There you have it, guys! We have successfully evaluated the expression −35×710+(−25+12)2-\frac{3}{5} \times \frac{7}{10}+\left(-\frac{2}{5}+\frac{1}{2}\right)^2, step-by-step. Remember, the key is to follow the order of operations (PEMDAS), break down the problem into smaller parts, and take your time. Don't worry if it seems challenging at first; the more you practice, the easier it becomes. Practice is paramount! Keep practicing these types of problems, and you'll find that evaluating expressions becomes second nature. Mastering expression evaluation not only improves your math skills but also boosts your confidence in tackling more complex problems. Every time you solve a problem, you are reinforcing the fundamental principles of mathematics. Keep up the great work, and happy calculating!