Evaluating |(1/2)(1^10 - 7 - 2^3)|: A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun little math problem that involves evaluating the absolute value of an expression. Specifically, we're going to break down how to solve |(1/2)(1^10 - 7 - 2^3)|. Don't worry, it's not as intimidating as it looks! We'll go through it step by step, making sure everyone can follow along. So, grab your pencils and let's get started!

Understanding the Expression

Before we jump into solving the problem, let's first understand what the expression is asking us to do. The expression |(1/2)(1^10 - 7 - 2^3)| contains several mathematical operations. The absolute value, denoted by the vertical bars ||, means that the final result will always be non-negative, regardless of whether the value inside the bars is positive or negative. We need to simplify the expression inside the absolute value first. This involves evaluating exponents, subtraction, and multiplication. By understanding the order of operations and what each symbol represents, we can approach the problem methodically and accurately. Each component plays a crucial role in determining the final outcome, and a clear understanding of these elements will help in solving similar mathematical problems in the future. Remember, math is all about breaking down complex problems into manageable steps!

Step-by-Step Evaluation

Okay, let's break down the expression step by step to make it super clear. First, we need to evaluate the exponents. We have 1^10 and 2^3. 1^10 simply means 1 multiplied by itself 10 times, which is still 1. So, 1^10 = 1. Next, we have 2^3, which means 2 multiplied by itself three times: 2 * 2 * 2 = 8. Now, let's rewrite the expression with these values: |(1/2)(1 - 7 - 8)|. Inside the parentheses, we have 1 - 7 - 8. Doing the subtraction from left to right, 1 - 7 equals -6. Then, -6 - 8 equals -14. So now our expression looks like this: |(1/2)(-14)|. Next, we multiply (1/2) by -14. Half of -14 is -7. So we have |-7|. Finally, we take the absolute value of -7. The absolute value of any number is its distance from zero, so the absolute value of -7 is 7. Therefore, |(1/2)(1^10 - 7 - 2^3)| = 7. See, that wasn't so bad! Each step is straightforward, and by breaking it down like this, it's easy to follow along and understand how we arrived at the final answer.

Detailed Breakdown of Each Operation

To ensure we've covered everything, let's delve deeper into each operation within the expression |(1/2)(1^10 - 7 - 2^3)|. Understanding the intricacies of each step is crucial for mastering similar mathematical problems. First, we tackled the exponents. The term 1^10 represents 1 raised to the power of 10. In simpler terms, it's 1 multiplied by itself ten times. No matter how many times you multiply 1 by itself, the result will always be 1. This is a fundamental property of the number 1 in exponentiation. Next, we evaluated 2^3, which is 2 raised to the power of 3. This means 2 multiplied by itself three times: 2 * 2 * 2 = 8. Understanding exponents is crucial for simplifying expressions and solving equations in algebra and beyond.

After handling the exponents, we moved on to the subtraction within the parentheses. We had 1 - 7 - 8. Subtraction is the inverse operation of addition, and it's essential to perform it accurately. Starting from left to right, 1 - 7 equals -6. Then, we subtracted 8 from -6, resulting in -14. The order of operations matters here; performing the subtractions in the correct sequence ensures we arrive at the correct intermediate result. Next, we multiplied (1/2) by -14. Multiplying a number by 1/2 is the same as dividing it by 2. Half of -14 is -7. Multiplication is a fundamental arithmetic operation, and understanding how to multiply fractions and negative numbers is crucial for solving mathematical problems.

Finally, we took the absolute value of -7. The absolute value of a number is its distance from zero on the number line, regardless of its sign. The absolute value of -7 is 7 because -7 is 7 units away from zero. The absolute value function is denoted by vertical bars ||, and it always returns a non-negative value. Understanding absolute values is essential for solving equations and inequalities in algebra and calculus.

Why Absolute Value Matters

So, why do we even care about absolute values? Great question! Absolute values are super important in math and have lots of real-world uses. Imagine you're measuring the distance between two points. Distance is always a positive value, right? You can't have a negative distance. That's where absolute value comes in handy. It ensures that we're always dealing with the magnitude or size of a number, regardless of its sign.

In mathematical terms, the absolute value function gives us the magnitude of a number. It's used extensively in calculus, analysis, and other advanced topics. For example, when dealing with limits and convergence, absolute values help us define how close a function or sequence gets to a certain value. They're also crucial in defining norms in vector spaces, which are used in linear algebra and functional analysis. In physics, absolute values can represent the magnitude of physical quantities like velocity or force. Even in computer science, absolute values are used in algorithms for error correction and data analysis.

Understanding absolute values helps you grasp more complex concepts and solve problems in various fields. So, next time you see those vertical bars, remember they're not just there to make things look complicated; they're ensuring we're dealing with positive magnitudes and making our calculations meaningful.

Common Mistakes to Avoid

Alright, let's chat about some common mistakes people often make when tackling problems like this. Recognizing these pitfalls can save you from unnecessary headaches and ensure you get the right answer every time.

One frequent error is messing up the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our problem, it's crucial to evaluate the exponents first before doing any subtraction. Another common mistake is miscalculating exponents. For instance, confusing 2^3 with 2 * 3 can lead to a completely wrong answer. Always remember that 2^3 means 2 multiplied by itself three times (2 * 2 * 2 = 8), not 2 multiplied by 3.

Sign errors are also very common, especially when dealing with negative numbers. Be extra careful when subtracting negative numbers or multiplying numbers with different signs. For example, 1 - 7 results in -6, not 6. Another mistake is forgetting to apply the absolute value at the end. The absolute value ensures that the final answer is non-negative. So, even if you end up with -7 inside the absolute value bars, the final answer should be 7. Finally, rushing through the problem without double-checking your work can lead to careless errors. Always take your time, show your steps, and review your calculations to avoid these common pitfalls. By being mindful of these mistakes, you can improve your accuracy and confidence in solving mathematical problems.

Practice Problems

To really nail this concept, let's try a few practice problems. These will help you get comfortable with evaluating expressions involving absolute values, exponents, and various arithmetic operations.

  1. |(1/3)(2^3 - 5)|
  2. |-2(3^2 - 10)|
  3. |(1/4)(4^2 - 2 * 6)|
  4. |5 - (1/2)(2^4)|
  5. |(1/5)(5^2 - 3 * 5)|

Take your time to solve these, and remember to follow the order of operations. By working through these problems, you'll reinforce your understanding and build confidence in your ability to tackle similar challenges.

Solutions to Practice Problems

Alright, let's check your answers to the practice problems! Here are the solutions, with a brief explanation for each one.

  1. |(1/3)(2^3 - 5)|: First, evaluate 2^3, which is 8. Then, the expression becomes |(1/3)(8 - 5)| = |(1/3)(3)| = |1| = 1.
  2. |-2(3^2 - 10)|: First, evaluate 3^2, which is 9. Then, the expression becomes |-2(9 - 10)| = |-2(-1)| = |2| = 2.
  3. |(1/4)(4^2 - 2 * 6)|: First, evaluate 4^2, which is 16, and 2 * 6, which is 12. Then, the expression becomes |(1/4)(16 - 12)| = |(1/4)(4)| = |1| = 1.
  4. |5 - (1/2)(2^4)|: First, evaluate 2^4, which is 16. Then, the expression becomes |5 - (1/2)(16)| = |5 - 8| = |-3| = 3.
  5. |(1/5)(5^2 - 3 * 5)|: First, evaluate 5^2, which is 25, and 3 * 5, which is 15. Then, the expression becomes |(1/5)(25 - 15)| = |(1/5)(10)| = |2| = 2.

How did you do? Hopefully, you nailed all of them! If you made any mistakes, don't worry. Just review the steps and try again. Practice makes perfect, and with a bit of effort, you'll be solving these types of problems like a pro!

Conclusion

So, there you have it! We've successfully evaluated the absolute value of the expression |(1/2)(1^10 - 7 - 2^3)|. By breaking down the problem into manageable steps and understanding the underlying principles, we were able to arrive at the solution with confidence. Remember, math is all about practice and perseverance. Keep honing your skills, and you'll be amazed at what you can achieve. Keep practicing, and you'll become a math whiz in no time! Keep up the great work, guys!