Solve The Following Mathematical Expressions Using The BODMAS Rule: a. [-10 - 3 × {4 - (-4)} × {3 - (5 - 3)}] b. [5 + 3{-36 - (16 - 14)} ÷ 3[17 + (-3) × 4 - 2 × (-7)]]

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In the realm of mathematics, the order of operations is paramount. The BODMAS acronym (Brackets, Orders, Division, Multiplication, Addition, and Subtraction) serves as a guiding principle for evaluating mathematical expressions accurately. This comprehensive guide delves into the intricacies of BODMAS, providing a step-by-step approach to solving complex expressions. We will dissect two challenging examples, demonstrating the application of BODMAS to arrive at the correct solutions.

Understanding the BODMAS Rule

The BODMAS rule is a fundamental concept in mathematics that dictates the sequence in which operations should be performed within an expression. Adhering to this rule ensures consistency and accuracy in mathematical calculations. Let's break down each component of the BODMAS acronym:

  • Brackets: Operations within brackets (parentheses), braces, and square brackets are always performed first. Start with the innermost brackets and work your way outwards.
  • Orders: This refers to powers, square roots, and other exponents. These operations are performed after brackets.
  • Division: Division operations are carried out before multiplication.
  • Multiplication: Multiplication operations are performed after division.
  • Addition: Addition operations are performed before subtraction.
  • Subtraction: Subtraction operations are carried out last.

By following the BODMAS rule meticulously, we can unravel complex expressions and arrive at the correct answers. It's a cornerstone of mathematical problem-solving, ensuring that calculations are performed in the correct order, leading to accurate results.

Example 1: Solving a Complex Expression with Multiple Brackets

Let's tackle our first expression:

a.

${-10 - 3 \times \{4 - (-4)\} \times \{3 - (5 - 3)\}}$

This expression presents a multi-layered challenge with nested brackets and a variety of operations. To solve it effectively, we must diligently apply the BODMAS rule, systematically working through each layer of complexity.

Step-by-Step Solution

  1. Innermost Brackets: Start by simplifying the expressions within the innermost brackets.

    • (5 - 3) = 2
    • The expression now becomes:
    ${-10 - 3 \times \{4 - (-4)\} \times \{3 - 2\}}$

  2. Curly Braces: Next, address the expressions within the curly braces.

    • For the first set of curly braces: 4 - (-4) = 4 + 4 = 8
    • For the second set of curly braces: 3 - 2 = 1
    • The expression simplifies to:
    ${-10 - 3 \times 8 \times 1}$

  3. Multiplication: Perform the multiplication operations.

    • 3 × 8 × 1 = 24
    • The expression now looks like this:
    ${-10 - 24}$

  4. Subtraction: Finally, perform the subtraction.

    • -10 - 24 = -34
    • Therefore, the solution to the expression is -34.

By meticulously following the BODMAS rule, we have successfully navigated the complexities of this expression, breaking it down into manageable steps and arriving at the accurate solution. Each step, from simplifying the innermost brackets to performing the final subtraction, was crucial in maintaining the integrity of the calculation.

Example 2: Solving an Expression with Division, Multiplication, and a Vinculum

Let's delve into our second expression:

b.

${5 + 3\{-36 - \overline{16 - 14}\} \div 3[17 + (-3) \times 4 - 2 \times (-7)]}$

This expression introduces a new element: the vinculum (the overline), which indicates that the expression beneath it should be treated as if it were enclosed in brackets. This adds another layer to the order of operations, making BODMAS even more critical.

Step-by-Step Solution

  1. Vinculum: Start by simplifying the expression under the vinculum.

    • 16 - 14 = 2
    • The expression now becomes:
    ${5 + 3\{-36 - 2\} \div 3[17 + (-3) \times 4 - 2 \times (-7)]}$

  2. Curly Braces: Address the expression within the curly braces.

    • -36 - 2 = -38
    • The expression simplifies to:
    ${5 + 3\{-38\} \div 3[17 + (-3) \times 4 - 2 \times (-7)]}$

  3. Square Brackets: Now, focus on the expression within the square brackets. This requires careful attention to the order of operations within the brackets.

    • Multiplication first: (-3) × 4 = -12 and 2 × (-7) = -14
    • The expression within the brackets becomes: 17 + (-12) - (-14)
    • Simplify further: 17 - 12 + 14 = 19
    • The entire expression now looks like this:
    ${5 + 3\{-38\} \div 3[19]}$

  4. Division: Perform the division operation.

    • First, multiply 3 with -38, that will be -114. Then the expression looks like this:
    ${5 + (-114) \div 3[19]}$
    • Next multiply 3 with 19, that will be 57. Then the expression looks like this:
    ${5 + (-114) \div 57}$
    • -114 ÷ 57 = -2
    • The expression simplifies to:
    ${5 + (-2)}$

  5. Addition: Finally, perform the addition.

    • 5 + (-2) = 3
    • Therefore, the solution to the expression is 3.

This example underscores the importance of meticulously following the BODMAS rule, especially when dealing with multiple operations and nested brackets. The vinculum added an extra layer of complexity, but by treating it as an implied bracket, we were able to navigate the expression systematically and arrive at the correct solution.

Conclusion: Mastering BODMAS for Mathematical Accuracy

The BODMAS rule is an indispensable tool in the realm of mathematics. By adhering to this order of operations, we can ensure accuracy and consistency in our calculations, even when dealing with complex expressions. The two examples we have explored demonstrate the power of BODMAS in simplifying intricate problems into manageable steps. From nested brackets to vinculums, BODMAS provides a clear roadmap for navigating the landscape of mathematical expressions.

Mastering BODMAS is not just about memorizing an acronym; it's about developing a deep understanding of the underlying principles of mathematical operations. It's about recognizing the hierarchy of operations and applying them in a logical, step-by-step manner. With practice and a solid grasp of BODMAS, you can confidently tackle any mathematical expression that comes your way, ensuring accurate and reliable results.

So, embrace the BODMAS rule, practice its application, and unlock your mathematical potential. It's the key to solving complex expressions with ease and precision, paving the way for success in mathematics and beyond.