Donte Simplified The Expression Below:$\[ \begin{array}{l} 4(1+3 I)-(8-5 I) \\ 4+3 I-8+5 I \\ -4+8 I \end{array} \\]What Mistake Did Donte Make?A. He Did Not Apply The Distributive Property Correctly For $4(1+3 I$\].B. He Did Not

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Introduction

Simplifying complex expressions is a crucial skill in mathematics, particularly in algebra and calculus. It requires careful application of mathematical operations, such as addition, subtraction, multiplication, and division. In this article, we will examine a complex expression that was simplified by Donte and identify the mistake he made.

The Complex Expression

The complex expression that Donte simplified is given by:

${ \begin{array}{l} 4(1+3 i)-(8-5 i) \\ 4+3 i-8+5 i \\ -4+8 i \end{array} \}$

Step-by-Step Simplification

To simplify this expression, we need to follow the order of operations (PEMDAS):

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

Let's apply these steps to the given expression:

  1. Parentheses: Evaluate the expressions inside the parentheses:

${ \begin{array}{l} 4(1+3 i) = 4 + 12 i \\ 8-5 i \end{array} \}$

  1. Exponents: There are no exponential expressions in this problem.

  2. Multiplication and Division: Evaluate any multiplication and division operations from left to right:

${ \begin{array}{l} 4 + 12 i - (8-5 i) \\ 4 + 12 i - 8 + 5 i \end{array} \}$

  1. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right:

${ \begin{array}{l} 4 + 12 i - 8 + 5 i \\ -4 + 17 i \end{array} \}$

Donte's Mistake

Donte's simplified expression is:

${ \begin{array}{l} -4+8 i \end{array} \}$

Comparing this with our step-by-step simplification, we can see that Donte made a mistake. The correct simplified expression is:

${ \begin{array}{l} -4+17 i \end{array} \}$

Conclusion

In conclusion, Donte made a mistake in simplifying the complex expression. He failed to apply the distributive property correctly for 4(1+3i)4(1+3 i). This resulted in an incorrect simplified expression. By following the order of operations and carefully applying mathematical operations, we can arrive at the correct simplified expression.

Common Mistakes in Simplifying Complex Expressions

There are several common mistakes that students make when simplifying complex expressions. Some of these mistakes include:

  • Failing to apply the distributive property: This is the mistake that Donte made. The distributive property states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac.
  • Not following the order of operations: This can lead to incorrect simplifications. It's essential to follow the order of operations (PEMDAS) to ensure that mathematical operations are evaluated correctly.
  • Not combining like terms: This can also lead to incorrect simplifications. Like terms are terms that have the same variable and exponent. Combining like terms involves adding or subtracting the coefficients of these terms.

Tips for Simplifying Complex Expressions

Simplifying complex expressions can be challenging, but with practice and patience, you can become proficient in it. Here are some tips to help you simplify complex expressions:

  • Read the problem carefully: Before starting to simplify the expression, read the problem carefully to understand what's being asked.
  • Apply the distributive property: The distributive property is a powerful tool for simplifying complex expressions. Make sure to apply it correctly.
  • Follow the order of operations: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be evaluated. Follow these rules to ensure that mathematical operations are evaluated correctly.
  • Combine like terms: Like terms are terms that have the same variable and exponent. Combining like terms involves adding or subtracting the coefficients of these terms.
  • Check your work: Finally, check your work to ensure that the simplified expression is correct.

Introduction

In our previous article, we examined a complex expression that was simplified by Donte and identified the mistake he made. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on simplifying complex expressions.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. This property allows us to distribute a single term to multiple terms inside a set of parentheses.

Q: Why is it essential to apply the distributive property when simplifying complex expressions?

A: Applying the distributive property is essential when simplifying complex expressions because it allows us to break down the expression into smaller, more manageable parts. This makes it easier to simplify the expression and arrive at the correct solution.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be evaluated. The acronym PEMDAS stands for:

  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

Q: Why is it essential to follow the order of operations when simplifying complex expressions?

A: Following the order of operations is essential when simplifying complex expressions because it ensures that mathematical operations are evaluated correctly. If we don't follow the order of operations, we may arrive at an incorrect solution.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x2x and 5x5x are like terms because they both have the variable xx and the same exponent (which is 1).

Q: How do I combine like terms?

A: To combine like terms, we add or subtract the coefficients of the terms. For example, if we have the expression 2x+5x2x + 5x, we can combine the like terms by adding the coefficients: 2+5=72 + 5 = 7. Therefore, the expression becomes 7x7x.

Q: What are some common mistakes to avoid when simplifying complex expressions?

A: Some common mistakes to avoid when simplifying complex expressions include:

  • Failing to apply the distributive property: This can lead to incorrect simplifications.
  • Not following the order of operations: This can also lead to incorrect simplifications.
  • Not combining like terms: This can also lead to incorrect simplifications.

Q: How can I practice simplifying complex expressions?

A: There are several ways to practice simplifying complex expressions, including:

  • Working through practice problems: Practice problems can help you develop your skills and build your confidence.
  • Using online resources: Online resources, such as video tutorials and practice exercises, can provide additional support and guidance.
  • Seeking help from a teacher or tutor: If you're struggling with simplifying complex expressions, don't hesitate to seek help from a teacher or tutor.

Conclusion

In conclusion, simplifying complex expressions requires careful attention to detail and a solid understanding of mathematical operations. By applying the distributive property, following the order of operations, and combining like terms, we can arrive at the correct solution. Remember to practice regularly and seek help when needed to become proficient in simplifying complex expressions.