Simplify The Expression:${ (-7 - 5) \div [-2 - 2 - (-6)] }$

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Introduction

Mathematical expressions can be complex and overwhelming, but with a clear understanding of the rules and operations, they can be simplified to reveal their underlying structure. In this article, we will focus on simplifying a specific mathematical expression, step by step, to demonstrate the process and provide a clear understanding of the concepts involved.

Understanding the Expression

The given expression is: ${ (-7 - 5) \div [-2 - 2 - (-6)] }$

This expression involves several mathematical operations, including subtraction, division, and negation. To simplify the expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Step 1: Evaluate Expressions Inside Parentheses

The first step is to evaluate the expressions inside the parentheses. We have two sets of parentheses:

  1. (−7−5)(-7 - 5)
  2. [−2−2−(−6)][-2 - 2 - (-6)]

Let's evaluate each of these expressions separately:

Evaluate (−7−5)(-7 - 5)

To evaluate this expression, we need to subtract 5 from -7:

(−7−5)=−12(-7 - 5) = -12

Evaluate [−2−2−(−6)][-2 - 2 - (-6)]

To evaluate this expression, we need to follow the order of operations:

  1. Subtract 2 from -2: −2−2=−4-2 - 2 = -4
  2. Add 6 to -4: −4+6=2-4 + 6 = 2

So, the value of the second set of parentheses is 2.

Step 2: Simplify the Expression

Now that we have evaluated the expressions inside the parentheses, we can simplify the original expression:

(−7−5)÷[−2−2−(−6)]=−12÷2(-7 - 5) \div [-2 - 2 - (-6)] = -12 \div 2

Step 3: Evaluate the Division

The final step is to evaluate the division operation:

−12÷2=−6-12 \div 2 = -6

Therefore, the simplified expression is:

−6-6

Conclusion

Simplifying mathematical expressions requires a clear understanding of the rules and operations involved. By following the order of operations and evaluating expressions inside parentheses, we can simplify complex expressions and reveal their underlying structure. In this article, we demonstrated the process of simplifying a specific mathematical expression, step by step, to provide a clear understanding of the concepts involved.

Frequently Asked Questions

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, you need to follow the order of operations and perform the operations inside the parentheses first.

Q: What is the difference between subtraction and negation?

A: Subtraction and negation are two different operations. Subtraction involves subtracting one number from another, while negation involves changing the sign of a number.

Additional Resources

For more information on simplifying mathematical expressions, check out the following resources:

  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Simplifying Expressions

Final Thoughts

Simplifying mathematical expressions is an essential skill that requires practice and patience. By following the order of operations and evaluating expressions inside parentheses, you can simplify complex expressions and reveal their underlying structure. Remember to always follow the rules and operations involved, and don't be afraid to ask for help when needed.