Step-by-Step Evaluation Of The Expression 14 + 3 × (34 - √(18 - 14))

In this comprehensive guide, we will meticulously evaluate the expression 14 + 3 × (34 - √(18 - 14)) ÷ 3 × [6 × 2 + 17 - (12 × 3)] step-by-step. We will break down each operation, explaining the order of operations and providing clear explanations for each simplification. This guide aims to enhance your understanding of mathematical expressions and order of operations, making complex calculations more approachable.

Stepwise Simplification: Unveiling the Solution

To accurately evaluate the expression, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures we perform the operations in the correct sequence, leading to the accurate result. Let's embark on this stepwise simplification journey.

1. Simplify within Parentheses and Brackets

Our initial focus lies within the parentheses and brackets. We'll begin with the innermost parentheses and work our way outwards.

• First Parentheses (18 - 14): Within the first set of parentheses, we have 18 - 14. This simple subtraction yields 4. The expression now transforms to: 14 + 3 × (34 - √4) ÷ 3 × [6 × 2 + 17 - (12 × 3)]

• Second Parentheses (34 - √4): Next, we encounter √4, which represents the square root of 4. The square root of 4 is 2. Substituting this value, we get (34 - 2), which simplifies to 32. The expression becomes: 14 + 3 × 32 ÷ 3 × [6 × 2 + 17 - (12 × 3)]

• Brackets [6 × 2 + 17 - (12 × 3)]: Now we address the brackets. Within the brackets, we have a combination of multiplication, addition, and subtraction. Following PEMDAS, we first perform the multiplications:

  • 6 × 2 = 12
  • 12 × 3 = 36 The expression within the brackets now looks like: [12 + 17 - 36]

Next, we perform the addition and subtraction from left to right:

• 12 + 17 = 29
• 29 - 36 = -7 Thus, the expression within the brackets simplifies to -7. The entire expression now reads: 14 + 3 × 32 ÷ 3 × (-7)

2. Perform Multiplication and Division (from left to right)

With the parentheses and brackets simplified, we move on to multiplication and division. These operations are performed from left to right.

• 3 × 32: Our first multiplication is 3 × 32, which equals 96. The expression becomes: 14 + 96 ÷ 3 × (-7)
• 96 ÷ 3: Next, we perform the division 96 ÷ 3, which results in 32. The expression is now: 14 + 32 × (-7)
• 32 × (-7): Finally, we multiply 32 × (-7), which equals -224. The expression is now: 14 + (-224)

3. Perform Addition and Subtraction (from left to right)

The final step involves addition and subtraction, again performed from left to right.

• 14 + (-224): We add 14 and -224, which is the same as subtracting 224 from 14. This gives us -210.

Therefore, the final simplified value of the expression is -210.

Breaking Down the Order of Operations (PEMDAS)

Understanding the order of operations is crucial for accurate mathematical calculations. The acronym PEMDAS serves as a helpful mnemonic:

• Parentheses: Operations within parentheses (and brackets) are performed first. Start with the innermost parentheses and work outwards.
• Exponents: Exponents (powers and roots) are evaluated next.
• Multiplication and Division: Multiplication and division are performed from left to right.
• Addition and Subtraction: Addition and subtraction are performed from left to right.

By consistently applying PEMDAS, you can confidently tackle complex mathematical expressions and arrive at the correct solution.

Common Pitfalls to Avoid

When evaluating expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Ignoring the Order of Operations: This is the most frequent error. Failing to follow PEMDAS can lead to drastically incorrect results.
• Incorrectly Simplifying Parentheses: Make sure to simplify expressions within parentheses completely before moving on.
• Sign Errors: Pay close attention to signs (positive and negative), especially when dealing with subtraction and multiplication of negative numbers.
• Calculation Mistakes: Double-check your arithmetic to avoid simple calculation errors.

By being mindful of these potential pitfalls, you can significantly improve your accuracy when evaluating mathematical expressions.

Example Walkthrough: Reinforcing the Process

Let's solidify our understanding with another example. Consider the expression: 2 × [15 - (3 + 2) × 2] + 8 ÷ 4

1. Parentheses (3 + 2): We begin with the innermost parentheses: 3 + 2 = 5. The expression becomes: 2 × [15 - 5 × 2] + 8 ÷ 4
2. Brackets [15 - 5 × 2]: Within the brackets, we perform multiplication first: 5 × 2 = 10. Then, subtraction: 15 - 10 = 5. The expression simplifies to: 2 × 5 + 8 ÷ 4
3. Multiplication and Division: We perform multiplication and division from left to right:
   • 2 × 5 = 10
   • 8 ÷ 4 = 2 The expression now is: 10 + 2
4. Addition: Finally, we add: 10 + 2 = 12

Therefore, the value of the expression 2 × [15 - (3 + 2) × 2] + 8 ÷ 4 is 12.

Practice Makes Perfect: Sharpening Your Skills

The key to mastering expression evaluation lies in consistent practice. Work through various examples, gradually increasing the complexity. Challenge yourself with expressions involving multiple parentheses, exponents, and a mix of operations. The more you practice, the more confident and proficient you'll become.

Conclusion: Mastering Mathematical Expressions

Evaluating mathematical expressions is a fundamental skill in mathematics and various fields. By understanding and consistently applying the order of operations (PEMDAS), you can confidently simplify even complex expressions. Remember to break down the problem into smaller, manageable steps, pay close attention to detail, and practice regularly. With dedication and the techniques outlined in this guide, you'll be well-equipped to conquer any mathematical expression that comes your way. This skill is not just about getting the right answer; it's about developing a logical and systematic approach to problem-solving, a valuable asset in academics and beyond.