Simplify The Given Expression Below: $ (1+2i) \cdot (5-3i) $A. $ 2-15i $ B. $ 3+2i $ C. $ 5-6i $ D. $ 11+7i $

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will focus on simplifying complex numbers, specifically the expression $(1+2i) \cdot (5-3i)$. We will break down the steps involved in simplifying this expression and provide a clear explanation of the process.

What are Complex Numbers?

Complex numbers are numbers that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. Complex numbers can be visualized as points on a coordinate plane, with the real part $a$ representing the x-coordinate and the imaginary part $b$ representing the y-coordinate.

Multiplying Complex Numbers

To simplify the expression $(1+2i) \cdot (5-3i)$, we need to multiply the two complex numbers. When multiplying complex numbers, we follow the distributive property, which states that for any complex numbers $a+bi$ and $c+di$, the product is given by:

$(a+bi) \cdot (c+di) = (ac-bd) + (ad+bc)i$

Step 1: Multiply the Real Parts

To simplify the expression $(1+2i) \cdot (5-3i)$, we start by multiplying the real parts of the two complex numbers:

$1 \cdot 5 = 5$

Step 2: Multiply the Imaginary Parts

Next, we multiply the imaginary parts of the two complex numbers:

$2i \cdot (-3i) = -6i^2$

Since $i^2 = -1$, we can substitute this value into the expression:

$-6i^2 = -6(-1) = 6$

Step 3: Combine the Real and Imaginary Parts

Now, we combine the real and imaginary parts of the expression:

$5 + 6 = 11$

$-6i^2 = 6$

So, the simplified expression is:

$11 + 6i$

However, we need to check if this is the correct answer among the options provided.

Checking the Options

Let's compare our simplified expression with the options provided:

A. $2-15i$ B. $3+2i$ C. $5-6i$ D. $11+7i$

Our simplified expression is $11+6i$, which is not among the options. However, we can see that option D is close, but it has a different imaginary part.

Conclusion

In conclusion, the correct answer is not among the options provided. However, we can see that the correct answer is $11+6i$, which is not among the options. This exercise highlights the importance of carefully following the steps involved in simplifying complex numbers.

Final Answer

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Introduction

In our previous article, we explored the concept of simplifying complex numbers, specifically the expression $(1+2i) \cdot (5-3i)$. We broke down the steps involved in simplifying this expression and provided a clear explanation of the process. In this article, we will continue to explore complex numbers and provide a Q&A guide to help you better understand this concept.

Q: What are complex numbers?

A: Complex numbers are numbers that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. Complex numbers can be visualized as points on a coordinate plane, with the real part $a$ representing the x-coordinate and the imaginary part $b$ representing the y-coordinate.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you follow the distributive property, which states that for any complex numbers $a+bi$ and $c+di$, the product is given by:

$(a+bi) \cdot (c+di) = (ac-bd) + (ad+bc)i$

Q: What is the difference between real and imaginary parts?

A: The real part of a complex number is the part that is not multiplied by $i$, while the imaginary part is the part that is multiplied by $i$. For example, in the complex number $3+4i$, the real part is $3$ and the imaginary part is $4i$.

Q: How do I simplify complex numbers?

A: To simplify complex numbers, you need to follow the steps involved in multiplying complex numbers. This includes multiplying the real parts, multiplying the imaginary parts, and combining the real and imaginary parts.

Q: What is the correct answer for the expression $(1+2i) \cdot (5-3i)$?

A: The correct answer for the expression $(1+2i) \cdot (5-3i)$ is $11+6i$. This is obtained by multiplying the real parts, multiplying the imaginary parts, and combining the real and imaginary parts.

Q: Why is it important to simplify complex numbers?

A: Simplifying complex numbers is important because it helps to:

• Reduce the complexity of expressions
• Make calculations easier
• Improve understanding of complex numbers
• Enhance problem-solving skills

Q: Can complex numbers be used in real-life applications?

A: Yes, complex numbers have numerous applications in various fields, including:

• Algebra
• Geometry
• Calculus
• Physics
• Engineering

Q: How do I visualize complex numbers?

A: Complex numbers can be visualized as points on a coordinate plane, with the real part representing the x-coordinate and the imaginary part representing the y-coordinate.

Conclusion

In conclusion, complex numbers are an essential concept in mathematics, and simplifying them is crucial for problem-solving and understanding complex numbers. We hope that this Q&A guide has helped you better understand complex numbers and how to simplify them.

Final Tips

• Practice simplifying complex numbers to improve your understanding and skills.
• Use visual aids, such as coordinate planes, to help you visualize complex numbers.
• Apply complex numbers to real-life applications to enhance your problem-solving skills.

Common Mistakes

• Failing to follow the distributive property when multiplying complex numbers.
• Not combining the real and imaginary parts correctly.
• Not visualizing complex numbers as points on a coordinate plane.

Additional Resources

• Khan Academy: Complex Numbers
• Mathway: Complex Numbers
• Wolfram Alpha: Complex Numbers

We hope that this Q&A guide has been helpful in your understanding of complex numbers and how to simplify them. If you have any further questions or need additional resources, please don't hesitate to ask.