Evaluating -5+(4)^2 ÷(3+1) ⋅ 3 Using Order Of Operations

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Evaluating mathematical expressions requires a systematic approach, especially when multiple operations are involved. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), provides a clear set of rules to ensure consistent and accurate results. This article will guide you through the step-by-step process of evaluating the expression $-5+(4)^2 \div(3+1) \cdot 3$, emphasizing the importance of following the correct order of operations.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before diving into the expression, let's briefly review the order of operations. PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), is a mnemonic device that helps us remember the correct sequence. In some regions, the acronym BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) is used, which conveys the same principles. Understanding and applying PEMDAS/BODMAS correctly is crucial for accurate mathematical calculations.

  • Parentheses/Brackets: Operations inside parentheses or brackets are performed first. This is to ensure that expressions grouped within these symbols are treated as a single entity.
  • Exponents/Orders: Next, we handle exponents or orders (powers and roots). Exponents indicate the number of times a base number is multiplied by itself.
  • Multiplication and Division: These operations are performed from left to right. It's important to note that multiplication and division have equal priority, so their order is determined by their position in the expression.
  • Addition and Subtraction: Similarly, addition and subtraction are performed from left to right. These operations also have equal priority and are executed in the order they appear.

By adhering to this order, we can break down complex expressions into manageable steps and arrive at the correct solution. Now, let's apply these principles to our expression.

Step-by-Step Evaluation of the Expression

To accurately evaluate the expression $-5+(4)^2 \div(3+1) \cdot 3$, we will meticulously follow the order of operations. Each step will be clearly explained to ensure a thorough understanding of the process.

1. Parentheses

The first step is to address the operation within the parentheses: $(3+1)$. This is a straightforward addition.

(3+1)=4(3+1) = 4

Now, we substitute this result back into the original expression:

−5+(4)2÷4⋅3-5 + (4)^2 \div 4 \cdot 3

By resolving the parentheses first, we simplify the expression and pave the way for the next operation.

2. Exponents

Next, we need to evaluate the exponent: $(4)^2$. This means 4 raised to the power of 2, which is 4 multiplied by itself.

(4)2=4⋅4=16(4)^2 = 4 \cdot 4 = 16

Substituting this back into the expression, we get:

−5+16÷4⋅3-5 + 16 \div 4 \cdot 3

With the parentheses and exponents resolved, we move on to the next level of operations: multiplication and division.

3. Multiplication and Division (from left to right)

Multiplication and division have equal priority, so we perform these operations from left to right. First, we encounter the division: $16 \div 4$.

16÷4=416 \div 4 = 4

Substituting this result, our expression becomes:

−5+4⋅3-5 + 4 \cdot 3

Now, we perform the multiplication: $4 \cdot 3$.

4⋅3=124 \cdot 3 = 12

Our expression is now simplified to:

−5+12-5 + 12

With multiplication and division completed, we are left with addition and subtraction.

4. Addition and Subtraction (from left to right)

Finally, we perform addition and subtraction from left to right. In this case, we have: $-5 + 12$.

−5+12=7-5 + 12 = 7

Therefore, the final result of the expression is 7.

Detailed Breakdown of the Solution

Let's recap the steps we took to evaluate the expression $-5+(4)^2 \div(3+1) \cdot 3$:

  1. Parentheses: $(3+1) = 4$. The expression becomes $-5 + (4)^2 \div 4 \cdot 3$.
  2. Exponents: $(4)^2 = 16$. The expression becomes $-5 + 16 \div 4 \cdot 3$.
  3. Division: $16 \div 4 = 4$. The expression becomes $-5 + 4 \cdot 3$.
  4. Multiplication: $4 \cdot 3 = 12$. The expression becomes $-5 + 12$.
  5. Addition: $-5 + 12 = 7$.

Thus, the final answer is 7. Each step was carefully executed according to the order of operations, ensuring the accuracy of the result.

Common Mistakes to Avoid

When evaluating mathematical expressions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help prevent errors and improve accuracy.

  • Ignoring the Order of Operations: The most common mistake is failing to follow the order of operations (PEMDAS/BODMAS). For instance, performing addition before multiplication or division can lead to a completely different result.
  • Incorrectly Handling Negative Signs: Negative signs can be tricky, especially when combined with exponents or other operations. It's crucial to pay close attention to the placement of negative signs and apply them correctly.
  • Misunderstanding Exponents: Exponents indicate repeated multiplication, not simple multiplication by the exponent itself. For example, $(4)^2$ means $4 \cdot 4$, not $4 \cdot 2$.
  • Not Working from Left to Right: For operations with equal priority (like multiplication and division, or addition and subtraction), it's essential to work from left to right. Failing to do so can alter the outcome.
  • Rushing Through the Steps: Mathematical errors often occur when rushing through the steps. Taking the time to carefully write out each step and double-check calculations can significantly reduce mistakes.

By being mindful of these common errors and diligently applying the order of operations, you can enhance your mathematical accuracy and problem-solving skills.

Practice Problems

To reinforce your understanding of the order of operations, let's explore a few practice problems. Working through these examples will solidify your skills and build confidence in evaluating mathematical expressions.

  1. Evaluate: $10 - 2 \cdot (3 + 1)$
  2. Evaluate: $5 + (16 \div 4) - 3^2$
  3. Evaluate: $20 \div (2 \cdot 5) + 6$

Solutions to Practice Problems

  1. Problem 1: $10 - 2 \cdot (3 + 1)$

    • Parentheses: $(3 + 1) = 4$. The expression becomes $10 - 2 \cdot 4$.
    • Multiplication: $2 \cdot 4 = 8$. The expression becomes $10 - 8$.
    • Subtraction: $10 - 8 = 2$.
    • Final Answer: 2

  2. Problem 2: $5 + (16 \div 4) - 3^2$

    • Parentheses: $(16 \div 4) = 4$. The expression becomes $5 + 4 - 3^2$.
    • Exponents: $3^2 = 9$. The expression becomes $5 + 4 - 9$.
    • Addition: $5 + 4 = 9$. The expression becomes $9 - 9$.
    • Subtraction: $9 - 9 = 0$.
    • Final Answer: 0

  3. Problem 3: $20 \div (2 \cdot 5) + 6$

    • Parentheses: $(2 \cdot 5) = 10$. The expression becomes $20 \div 10 + 6$.
    • Division: $20 \div 10 = 2$. The expression becomes $2 + 6$.
    • Addition: $2 + 6 = 8$.
    • Final Answer: 8

By practicing these problems and carefully applying the order of operations, you can become proficient in evaluating complex mathematical expressions.

Conclusion

In conclusion, evaluating the expression $-5+(4)^2 \div(3+1) \cdot 3$ demonstrates the critical importance of following the order of operations (PEMDAS/BODMAS). By systematically addressing parentheses, exponents, multiplication and division, and finally, addition and subtraction, we arrived at the correct answer of 7. Understanding and applying these rules not only ensures accurate calculations but also builds a strong foundation for more advanced mathematical concepts. Remember to take your time, break down complex expressions into manageable steps, and double-check your work to avoid common mistakes. With consistent practice and a clear understanding of PEMDAS/BODMAS, you can confidently tackle a wide range of mathematical problems.