Select The Correct Answer.Which Complex Number Is Equivalent To This Expression? $\[\frac{1}{3}(6+3i) - \frac{2}{3}(6-12i)\\]A. \[$6-7i\$\] B. \[$-2-7i\$\] C. \[$-2+9i\$\] D. \[$6+9i\$\]

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Introduction

Complex numbers are an extension of the real number system, which includes imaginary numbers. They are used to represent points in a two-dimensional plane and have numerous applications in mathematics, science, and engineering. In this article, we will focus on solving complex number expressions, specifically the given expression ${\frac{1}{3}(6+3i) - \frac{2}{3}(6-12i)}$.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit, which satisfies the equation i2=−1i^2 = -1. The real part of a complex number is aa, and the imaginary part is bb. Complex numbers can be added, subtracted, multiplied, and divided just like real numbers.

Solving the Given Expression

To solve the given expression, we need to follow the order of operations (PEMDAS):

  1. Multiply the terms inside the parentheses by the corresponding fractions.
  2. Combine like terms.

Let's break down the expression:

13(6+3i)−23(6−12i){\frac{1}{3}(6+3i) - \frac{2}{3}(6-12i)}

First, multiply the terms inside the parentheses by the corresponding fractions:

13(6+3i)=2+i{\frac{1}{3}(6+3i) = 2 + i}

23(6−12i)=−4+8i{\frac{2}{3}(6-12i) = -4 + 8i}

Now, subtract the second term from the first term:

(2+i)−(−4+8i){(2 + i) - (-4 + 8i)}

Combine like terms:

2+i+4−8i{2 + i + 4 - 8i}

6−7i{6 - 7i}

Comparing the Result with the Options

Now that we have solved the expression, let's compare the result with the given options:

A. ${6-7i}$ B. ${-2-7i}$ C. ${-2+9i}$ D. ${6+9i}$

The result we obtained, ${6-7i}$, matches option A.

Conclusion

In this article, we solved a complex number expression using the order of operations (PEMDAS) and compared the result with the given options. We found that the correct answer is option A, ${6-7i}$. This exercise demonstrates the importance of following the order of operations and combining like terms when solving complex number expressions.

Additional Tips and Tricks

When solving complex number expressions, remember to:

  • Follow the order of operations (PEMDAS)
  • Combine like terms
  • Use the distributive property to multiply terms inside parentheses
  • Simplify the expression by combining like terms

By following these tips and tricks, you can become proficient in solving complex number expressions and tackle more challenging problems with confidence.

Common Mistakes to Avoid

When solving complex number expressions, avoid the following common mistakes:

  • Not following the order of operations (PEMDAS)
  • Not combining like terms
  • Not using the distributive property to multiply terms inside parentheses
  • Not simplifying the expression by combining like terms

By being aware of these common mistakes, you can avoid them and produce accurate results.

Real-World Applications

Complex numbers have numerous real-world applications, including:

  • Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: Complex numbers are used to represent signals and analyze their frequency content.
  • Control systems: Complex numbers are used to represent the behavior of control systems and analyze their stability.

By understanding complex numbers and how to solve complex number expressions, you can apply these concepts to real-world problems and make a meaningful impact in various fields.

Final Thoughts

Introduction

In our previous article, we solved a complex number expression using the order of operations (PEMDAS) and compared the result with the given options. In this article, we will provide a Q&A guide to help you better understand complex number expressions and how to solve them.

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit, which satisfies the equation i2=−1i^2 = -1.

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

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for:

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

Q: How do I simplify a complex number expression?

A: To simplify a complex number expression, follow these steps:

  1. Multiply the terms inside parentheses by the corresponding fractions.
  2. Combine like terms.
  3. Simplify the expression by combining like terms.

Q: What is the distributive property?

A: The distributive property is a rule that allows you to multiply a single term by multiple terms inside parentheses. It states that:

a(b+c)=ab+aca(b + c) = ab + ac

Q: How do I use the distributive property to simplify a complex number expression?

A: To use the distributive property to simplify a complex number expression, follow these steps:

  1. Multiply the term outside the parentheses by each term inside the parentheses.
  2. Combine like terms.
  3. Simplify the expression by combining like terms.

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

A: Some common mistakes to avoid when solving complex number expressions include:

  • Not following the order of operations (PEMDAS)
  • Not combining like terms
  • Not using the distributive property to multiply terms inside parentheses
  • Not simplifying the expression by combining like terms

Q: What are some real-world applications of complex numbers?

A: Complex numbers have numerous real-world applications, including:

  • Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: Complex numbers are used to represent signals and analyze their frequency content.
  • Control systems: Complex numbers are used to represent the behavior of control systems and analyze their stability.

Q: How can I practice solving complex number expressions?

A: To practice solving complex number expressions, try the following:

  • Start with simple expressions and gradually move on to more complex ones.
  • Use online resources, such as practice problems and worksheets.
  • Work with a partner or join a study group to practice solving complex number expressions together.

Conclusion

In this Q&A guide, we provided answers to common questions about complex number expressions and how to solve them. By following the tips and tricks outlined in this article, you can become proficient in solving complex number expressions and tackle more challenging problems with confidence. Remember to avoid common mistakes and apply complex numbers to real-world problems to make a meaningful impact.