Multiply And Simplify The Product \[$(8-5i)^2\$\].Select The Product.A. 39 B. 89 C. \[$39 - 80i\$\] D. \[$89 - 80i\$\]

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Introduction

In mathematics, 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 are used to represent points in a two-dimensional plane, and they have numerous applications in mathematics, science, and engineering. In this article, we will focus on multiplying and simplifying complex numbers, specifically the product of (8-5i)^2.

Multiplying Complex Numbers

To multiply complex numbers, we can use the distributive property and the fact that i^2 = -1. Let's consider the product (8-5i)^2. We can expand this product using the distributive property:

(8-5i)^2 = (8-5i)(8-5i)

Using the distributive property, we get:

(8-5i)(8-5i) = 8(8-5i) - 5i(8-5i)

Expanding further, we get:

8(8-5i) - 5i(8-5i) = 64 - 40i - 40i + 25i^2

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

64 - 40i - 40i + 25i^2 = 64 - 80i - 25

Combining like terms, we get:

64 - 80i - 25 = 39 - 80i

Simplifying the Product

Now that we have expanded and simplified the product (8-5i)^2, we can see that it is equal to 39 - 80i. This is the final answer.

Conclusion

In this article, we have multiplied and simplified the complex number (8-5i)^2. We used the distributive property and the fact that i^2 = -1 to expand and simplify the product. The final answer is 39 - 80i.

Answer

The correct answer is:

  • C. [39 - 80i]

Key Takeaways

  • To multiply complex numbers, we can use the distributive property and the fact that i^2 = -1.
  • When expanding and simplifying complex numbers, we need to combine like terms and use the fact that i^2 = -1.
  • The final answer is 39 - 80i.

Practice Problems

  1. Multiply and simplify the complex number (3+4i)^2.
  2. Multiply and simplify the complex number (2-3i)(4+5i).
  3. Multiply and simplify the complex number (1+2i)^3.

Solutions

  1. (3+4i)^2 = 9 + 24i + 16i^2 = 9 + 24i - 16 = -7 + 24i
  2. (2-3i)(4+5i) = 8 + 10i - 12i - 15i^2 = 8 - 2i + 15 = 23 - 2i
  3. (1+2i)^3 = (1+2i)(1+2i)(1+2i) = (1+4i+4i^2)(1+2i) = (-3+4i)(1+2i) = -3 - 6i + 4i + 8i^2 = -3 - 2i - 8 = -11 - 2i
    Multiplying and Simplifying Complex Numbers: Q&A =====================================================

Introduction

In our previous article, we explored the concept of multiplying and simplifying complex numbers. We learned how to use the distributive property and the fact that i^2 = -1 to expand and simplify complex numbers. In this article, we will answer some frequently asked questions about multiplying and simplifying complex numbers.

Q&A

Q: What is the difference between multiplying complex numbers and multiplying real numbers?

A: When multiplying complex numbers, we need to use the distributive property and the fact that i^2 = -1. This is different from multiplying real numbers, where we can simply multiply the numbers together.

Q: How do I know when to use the distributive property when multiplying complex numbers?

A: You should use the distributive property when multiplying complex numbers whenever you see a product of two or more complex numbers. This will help you to expand and simplify the product.

Q: What is the value of i^2?

A: The value of i^2 is -1. This is a fundamental property of complex numbers that we use when multiplying and simplifying complex numbers.

Q: Can I multiply complex numbers with different powers of i?

A: Yes, you can multiply complex numbers with different powers of i. For example, you can multiply (3+4i) with (2-3i) and (1+2i) with (4+5i).

Q: How do I simplify complex numbers with multiple terms?

A: To simplify complex numbers with multiple terms, you should combine like terms and use the fact that i^2 = -1. For example, if you have the expression 2i + 3i^2, you can simplify it by combining like terms and using the fact that i^2 = -1.

Q: Can I multiply complex numbers with real numbers?

A: Yes, you can multiply complex numbers with real numbers. For example, you can multiply (3+4i) with 2 or (1+2i) with 3.

Q: How do I know when to use the fact that i^2 = -1 when multiplying complex numbers?

A: You should use the fact that i^2 = -1 whenever you see a product of two or more complex numbers that contains i^2. This will help you to simplify the product.

Q: Can I multiply complex numbers with negative coefficients?

A: Yes, you can multiply complex numbers with negative coefficients. For example, you can multiply (-3+4i) with (-2-3i).

Q: How do I simplify complex numbers with negative coefficients?

A: To simplify complex numbers with negative coefficients, you should combine like terms and use the fact that i^2 = -1. For example, if you have the expression -2i - 3i^2, you can simplify it by combining like terms and using the fact that i^2 = -1.

Conclusion

In this article, we have answered some frequently asked questions about multiplying and simplifying complex numbers. We have covered topics such as the difference between multiplying complex numbers and real numbers, how to use the distributive property, and how to simplify complex numbers with multiple terms. We hope that this article has been helpful in clarifying any questions you may have had about multiplying and simplifying complex numbers.

Practice Problems

  1. Multiply and simplify the complex number (2+3i)(4-5i).
  2. Multiply and simplify the complex number (1-2i)^2.
  3. Multiply and simplify the complex number (3+4i)(2-3i).

Solutions

  1. (2+3i)(4-5i) = 8 - 10i + 12i - 15i^2 = 8 + 2i + 15 = 23 + 2i
  2. (1-2i)^2 = 1 - 4i + 4i^2 = 1 - 4i - 4 = -3 - 4i
  3. (3+4i)(2-3i) = 6 - 9i + 8i - 12i^2 = 6 - i + 12 = 18 - i