Calculate The Following Respecting The Order Of Operations And The Use Of Parentheses: a) 8 + 18 : 2 + 8 × 3 + 8 : 4 × 5 b) 10 000 : (100 × 10 + 100 : 10 - 10 : 1) × 80 c) 7 + [7 + 7 × (7 + 7 × 7 - 7 × 7)] : 7

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This article aims to provide a comprehensive guide on how to calculate mathematical expressions accurately by adhering to the order of operations and the rules governing the use of parentheses. Mastering these principles is fundamental for success in mathematics and related fields. We'll break down the concepts step-by-step and illustrate them with detailed examples, ensuring a clear understanding for learners of all levels.

Understanding the Order of Operations

The order of operations, often remembered by the acronym PEMDAS or BODMAS, dictates the sequence in which mathematical operations should be performed. This universal convention ensures that everyone arrives at the same answer when evaluating an expression. Ignoring this order can lead to incorrect results, emphasizing the importance of adhering to it strictly.

The acronyms stand for:

  • Parentheses (or Brackets)
  • Exponents (or Orders)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Let's delve deeper into each component:

  1. Parentheses/Brackets: Operations enclosed within parentheses or brackets are always performed first. This allows us to group parts of an expression and prioritize their evaluation. For instance, in the expression 2 × (3 + 4), we first calculate 3 + 4 = 7 and then multiply by 2, resulting in 14.

  2. Exponents/Orders: Exponents (or orders, powers, and roots) are evaluated next. An exponent indicates how many times a base number is multiplied by itself. For example, in 5^2 (5 squared), 5 is the base and 2 is the exponent, meaning 5 is multiplied by itself twice (5 × 5 = 25).

  3. Multiplication and Division: These operations have equal priority and are performed from left to right. This means that if both multiplication and division are present in an expression, we evaluate them in the order they appear from left to right. For example, in 10 ÷ 2 × 3, we first divide 10 by 2 (resulting in 5) and then multiply by 3, yielding 15.

  4. Addition and Subtraction: Similar to multiplication and division, addition and subtraction have equal priority and are performed from left to right. In the expression 8 - 3 + 2, we first subtract 3 from 8 (resulting in 5) and then add 2, giving us 7.

By diligently following PEMDAS/BODMAS, we can ensure accurate and consistent calculations, regardless of the complexity of the expression. This foundation is crucial for tackling more advanced mathematical concepts.

The Significance of Parentheses

Parentheses play a crucial role in mathematical expressions. Parentheses are essential tools that dictate the order in which operations are performed. They act as containers, grouping parts of an expression and signaling that these enclosed operations should be evaluated before any others. This capability to override the standard order of operations is what makes parentheses so powerful. By strategically placing parentheses, we can change the entire outcome of a calculation.

Consider the simple expression 5 + 3 × 2. If we follow the order of operations without parentheses, we first perform the multiplication 3 × 2 = 6 and then add 5, resulting in 5 + 6 = 11. However, if we introduce parentheses and write (5 + 3) × 2, we first evaluate the expression within the parentheses, 5 + 3 = 8, and then multiply by 2, yielding 8 × 2 = 16. The difference between 11 and 16 highlights the significant impact parentheses can have.

The strategic use of parentheses allows us to express complex relationships and calculations in a clear and unambiguous manner. In more intricate expressions, multiple sets of parentheses may be nested within each other. When dealing with nested parentheses, we work from the innermost set outwards. This ensures that the operations are performed in the correct sequence, leading to the accurate result.

For example, let's analyze the expression 2 × [3 + (4 × 5)]. Here, we have brackets (which function similarly to parentheses) enclosing another set of parentheses. Following the rule of innermost outwards, we first calculate 4 × 5 = 20. Then, we substitute this result back into the expression, giving us 2 × [3 + 20]. Next, we evaluate the expression within the brackets, 3 + 20 = 23. Finally, we multiply by 2, resulting in 2 × 23 = 46.

Understanding how to use parentheses effectively is a cornerstone of mathematical proficiency. It empowers us to construct and interpret complex expressions with precision, ensuring that our calculations accurately reflect the intended relationships between numbers and operations. It's a skill that's invaluable not just in mathematics but also in various fields that rely on logical and quantitative reasoning.

Solving the Problems

Now, let's apply our understanding of the order of operations and parentheses to solve the given problems:

a) 8 + 18 : 2 + 8 × 3 + 8 : 4 × 5

To solve this expression, we will follow the PEMDAS/BODMAS order of operations. Since there are no parentheses or exponents, we proceed with multiplication and division from left to right, followed by addition and subtraction from left to right.

  1. Division: 18 : 2 = 9 and 8 : 4 = 2

    The expression becomes: 8 + 9 + 8 × 3 + 2 × 5

  2. Multiplication: 8 × 3 = 24 and 2 × 5 = 10

    The expression becomes: 8 + 9 + 24 + 10

  3. Addition: Now, we perform addition from left to right:

    • 8 + 9 = 17
    • 17 + 24 = 41
    • 41 + 10 = 51

Therefore, the solution to the expression 8 + 18 : 2 + 8 × 3 + 8 : 4 × 5 is 51.

b) 10 000 : (100 × 10 + 100 : 10 - 10 : 1) × 80

This expression includes parentheses, so we must evaluate the expression within them first, adhering to PEMDAS/BODMAS within the parentheses as well.

  1. Parentheses: We focus on the expression inside the parentheses: (100 × 10 + 100 : 10 - 10 : 1)

    • Multiplication: 100 × 10 = 1000
    • Division: 100 : 10 = 10 and 10 : 1 = 10

    The expression inside the parentheses becomes: 1000 + 10 - 10

    • Addition and Subtraction (from left to right):
      • 1000 + 10 = 1010
      • 1010 - 10 = 1000

    So, the expression within the parentheses simplifies to 1000.

    Now, the original expression becomes: 10 000 : 1000 × 80

  2. Division: 10 000 : 1000 = 10

    The expression becomes: 10 × 80

  3. Multiplication: 10 × 80 = 800

Therefore, the solution to the expression 10 000 : (100 × 10 + 100 : 10 - 10 : 1) × 80 is 800.

c) 7 + [7 + 7 × (7 + 7 × 7 - 7 × 7)] : 7

This expression involves nested parentheses (brackets and parentheses), so we will work from the innermost set outwards, following PEMDAS/BODMAS at each level.

  1. Innermost Parentheses: We focus on the expression inside the parentheses: (7 + 7 × 7 - 7 × 7)

    • Multiplication: 7 × 7 = 49 (occurs twice)

    The expression becomes: (7 + 49 - 49)

    • Addition and Subtraction (from left to right):
      • 7 + 49 = 56
      • 56 - 49 = 7

    So, the expression within the innermost parentheses simplifies to 7.

    Now, the expression becomes: 7 + [7 + 7 × 7] : 7

  2. Brackets: We focus on the expression inside the brackets: [7 + 7 × 7]

    • Multiplication: 7 × 7 = 49

    The expression becomes: [7 + 49]

    • Addition: 7 + 49 = 56

    So, the expression within the brackets simplifies to 56.

    Now, the expression becomes: 7 + 56 : 7

  3. Division: 56 : 7 = 8

    The expression becomes: 7 + 8

  4. Addition: 7 + 8 = 15

Therefore, the solution to the expression 7 + [7 + 7 × (7 + 7 × 7 - 7 × 7)] : 7 is 15.

Conclusion

By consistently applying the order of operations and understanding the role of parentheses, you can confidently tackle complex mathematical expressions. Remember the PEMDAS/BODMAS rule and work from the innermost parentheses outwards when dealing with nested structures. Practice is key to mastering these concepts, so continue to solve various problems to solidify your understanding. This will not only enhance your mathematical skills but also your problem-solving abilities in general.