Simplify: ∣ 3 − 6 ∣ − ( 12 ÷ 6 + 1 ) 2 |3-6|-(12 \div 6+1)^2 ∣3 − 6∣ − ( 12 ÷ 6 + 1 ) 2 A. -2 B. -12 C. -6 D. 0

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Understanding the Problem

The given expression involves absolute value, division, and exponentiation. To simplify it, we need to follow the order of operations (PEMDAS/BODMAS), which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.

Breaking Down the Expression

Let's break down the given expression into smaller parts:

  1. ∣3−6∣|3-6|
  2. 12÷612 \div 6
  3. 12÷6+112 \div 6 + 1
  4. (12÷6+1)2(12 \div 6 + 1)^2

Evaluating the Absolute Value

The absolute value of a number is its distance from zero on the number line. In this case, we have ∣3−6∣|3-6|, which can be evaluated as follows:

∣3−6∣=∣−3∣=3|3-6| = |-3| = 3

Evaluating the Division

Next, we need to evaluate the division 12÷612 \div 6:

12÷6=212 \div 6 = 2

Evaluating the Expression Inside the Parentheses

Now, we need to evaluate the expression inside the parentheses: 12÷6+112 \div 6 + 1:

12÷6+1=2+1=312 \div 6 + 1 = 2 + 1 = 3

Evaluating the Exponentiation

Finally, we need to evaluate the exponentiation (12÷6+1)2(12 \div 6 + 1)^2:

(12÷6+1)2=32=9(12 \div 6 + 1)^2 = 3^2 = 9

Putting it All Together

Now that we have evaluated all the parts, we can put them together to simplify the original expression:

∣3−6∣−(12÷6+1)2=3−9=−6|3-6|-(12 \div 6+1)^2 = 3 - 9 = -6

Conclusion

Therefore, the simplified value of the given expression is −6-6. This is the correct answer.

Why is this the Correct Answer?

To understand why this is the correct answer, let's analyze the expression step by step. We started by evaluating the absolute value, which gave us 33. Then, we evaluated the division, which gave us 22. Next, we evaluated the expression inside the parentheses, which gave us 33. Finally, we evaluated the exponentiation, which gave us 99. When we put it all together, we got 3−9=−63 - 9 = -6.

What if we had chosen a different order of operations?

If we had chosen a different order of operations, we might have gotten a different answer. However, the order of operations is a well-established rule in mathematics, and it ensures that we evaluate expressions in a consistent and predictable way.

What are the implications of this result?

The result of this expression has implications for various fields, including mathematics, science, and engineering. For example, in mathematics, this result can be used to simplify complex expressions and solve equations. In science, this result can be used to model real-world phenomena and make predictions. In engineering, this result can be used to design and optimize systems.

What are the limitations of this result?

While this result is correct, it is not without limitations. For example, this result assumes that the expression is well-defined and that the order of operations is followed. In some cases, the expression may be undefined or the order of operations may be ambiguous, which can lead to different results.

What are the future directions of this research?

The research on simplifying expressions like this one is ongoing, and there are many future directions to explore. For example, researchers can investigate new methods for simplifying expressions, develop new algorithms for evaluating expressions, and apply these results to real-world problems.

Conclusion

In conclusion, the simplified value of the given expression is −6-6. This result has implications for various fields, including mathematics, science, and engineering. While this result is correct, it is not without limitations, and there are many future directions to explore.

Final Answer

The final answer is: −6\boxed{-6}

Frequently Asked Questions

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is often remembered using the acronym PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.

Q: Why is it important to follow the order of operations?

A: Following the order of operations is important because it ensures that we evaluate expressions in a consistent and predictable way. If we don't follow the order of operations, we may get different answers, which can lead to errors and confusion.

Q: How do I evaluate an absolute value expression?

A: To evaluate an absolute value expression, we need to find the distance of the number inside the absolute value bars from zero on the number line. If the number is positive, the absolute value is the same as the number. If the number is negative, the absolute value is the opposite of the number.

Q: How do I evaluate an exponentiation expression?

A: To evaluate an exponentiation expression, we need to raise the number inside the parentheses to the power of the exponent. For example, 23=82^3 = 8.

Q: How do I evaluate a division expression?

A: To evaluate a division expression, we need to divide the number on the top by the number on the bottom. For example, 12÷6=212 \div 6 = 2.

Q: How do I evaluate an addition and subtraction expression?

A: To evaluate an addition and subtraction expression, we need to add or subtract the numbers from left to right. For example, 3+2−1=43 + 2 - 1 = 4.

Q: What if I get a different answer than the one in the solution?

A: If you get a different answer than the one in the solution, it's possible that you made a mistake in your calculations. Double-check your work and make sure you followed the order of operations correctly.

Q: Can I use a calculator to evaluate the expression?

A: Yes, you can use a calculator to evaluate the expression. However, keep in mind that calculators may have different order of operations or rounding errors, which can affect the accuracy of the result.

Q: How do I apply this result to real-world problems?

A: This result can be applied to various real-world problems, such as modeling population growth, predicting stock prices, or designing electrical circuits. The key is to identify the mathematical relationships between the variables and use the order of operations to evaluate the expressions.

Q: What are some common mistakes to avoid when evaluating expressions?

A: Some common mistakes to avoid when evaluating expressions include:

  • Not following the order of operations
  • Not evaluating expressions inside parentheses first
  • Not using the correct order of operations for exponents, multiplication and division, and addition and subtraction
  • Not checking for rounding errors or calculator errors

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you can use various techniques such as factoring, combining like terms, or using the distributive property. You can also use algebraic manipulations such as adding or subtracting the same value to both sides of an equation.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, such as:

  • Modeling population growth and predicting future population sizes
  • Predicting stock prices and making investment decisions
  • Designing electrical circuits and predicting circuit behavior
  • Solving optimization problems and finding the maximum or minimum value of a function

Q: How do I evaluate expressions with multiple variables?

A: To evaluate expressions with multiple variables, you need to follow the order of operations and use the correct order of operations for each operation. You can also use algebraic manipulations such as substituting values or using the distributive property.

Q: What are some common pitfalls to avoid when evaluating expressions with multiple variables?

A: Some common pitfalls to avoid when evaluating expressions with multiple variables include:

  • Not following the order of operations
  • Not using the correct order of operations for each operation
  • Not checking for rounding errors or calculator errors
  • Not considering the relationships between the variables

Q: How do I apply this result to optimization problems?

A: This result can be applied to optimization problems by using the order of operations to evaluate the expressions and find the maximum or minimum value of a function. You can also use algebraic manipulations such as adding or subtracting the same value to both sides of an equation.

Q: What are some real-world applications of optimization problems?

A: Optimization problems have many real-world applications, such as:

  • Finding the minimum cost of producing a product
  • Finding the maximum profit of a business
  • Designing the most efficient supply chain
  • Finding the optimal solution to a complex problem

Q: How do I evaluate expressions with absolute values and exponents?

A: To evaluate expressions with absolute values and exponents, you need to follow the order of operations and use the correct order of operations for each operation. You can also use algebraic manipulations such as substituting values or using the distributive property.

Q: What are some common pitfalls to avoid when evaluating expressions with absolute values and exponents?

A: Some common pitfalls to avoid when evaluating expressions with absolute values and exponents include:

  • Not following the order of operations
  • Not using the correct order of operations for each operation
  • Not checking for rounding errors or calculator errors
  • Not considering the relationships between the variables

Q: How do I apply this result to real-world problems involving absolute values and exponents?

A: This result can be applied to real-world problems involving absolute values and exponents by using the order of operations to evaluate the expressions and find the maximum or minimum value of a function. You can also use algebraic manipulations such as adding or subtracting the same value to both sides of an equation.

Q: What are some real-world applications of absolute values and exponents?

A: Absolute values and exponents have many real-world applications, such as:

  • Modeling population growth and predicting future population sizes
  • Predicting stock prices and making investment decisions
  • Designing electrical circuits and predicting circuit behavior
  • Solving optimization problems and finding the maximum or minimum value of a function