Multiply The Following Complex Numbers: $(3-5i)(5-7i$\]A. $40+46i$ B. $-20-46i$ C. $-20+46i$ D. $40-46i$

Introduction

Complex numbers are mathematical expressions that consist of a real number and an imaginary number. They are used to represent points in a two-dimensional plane and are essential in various fields, including mathematics, physics, and engineering. In this article, we will focus on multiplying complex numbers, which is a fundamental operation in complex number arithmetic.

What are Complex Numbers?

A complex number is a number 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 $i^2 = -1$. The real number $a$ is called the real part, and the imaginary number $bi$ is called the imaginary part.

Multiplying Complex Numbers

To multiply complex numbers, we use the distributive property of multiplication over addition. Let's consider two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, where $a$, $b$, $c$, and $d$ are real numbers. The product of $z_1$ and $z_2$ is given by:

$z_1z_2 = (a + bi)(c + di)$

Using the distributive property, we can expand the product as follows:

$z_1z_2 = ac + adi + bci + bdi^2$

Since $i^2 = -1$, we can simplify the expression as:

$z_1z_2 = ac + adi + bci - bd$

Combining like terms, we get:

$z_1z_2 = (ac - bd) + (ad + bc)i$

Example: Multiplying Complex Numbers

Let's consider the complex numbers $(3-5i)$ and $(5-7i)$. We want to multiply these two complex numbers using the formula we derived earlier.

$(3-5i)(5-7i) = (3 \cdot 5 - (-5) \cdot (-7)) + (3 \cdot (-7) + (-5) \cdot 5)i$

Simplifying the expression, we get:

$(3-5i)(5-7i) = (15 - 35) + (-21 - 25)i$

Combining like terms, we get:

$(3-5i)(5-7i) = -20 - 46i$

Conclusion

Multiplying complex numbers is a fundamental operation in complex number arithmetic. By using the distributive property of multiplication over addition, we can multiply complex numbers and simplify the resulting expression. In this article, we derived the formula for multiplying complex numbers and applied it to an example. We hope this article has provided a clear understanding of how to multiply complex numbers.

Answer

The correct answer is:

• B. $-20-46i$

Additional Examples

Here are a few more examples of multiplying complex numbers:

• $(2+3i)(4+5i) = (8 + 10i + 12i + 15i^2) = (8 + 22i - 15) = -7 + 22i$
• $(1-2i)(3+4i) = (3 + 4i - 6i - 8i^2) = (3 - 2i + 8) = 11 - 2i$
• $(4+5i)(2-3i) = (8 - 12i + 10i - 15i^2) = (8 - 2i + 15) = 23 - 2i$

Introduction

In our previous article, we discussed how to multiply complex numbers using the distributive property of multiplication over addition. We also provided examples of multiplying complex numbers and derived the formula for multiplying complex numbers. In this article, we will answer some frequently asked questions about multiplying complex numbers.

Q: What is the formula for multiplying complex numbers?

A: The formula for multiplying complex numbers is:

$z_1z_2 = (a + bi)(c + di) = (ac - bd) + (ad + bc)i$

where $a$, $b$, $c$, and $d$ are real numbers.

Q: How do I multiply complex numbers with different signs?

A: When multiplying complex numbers with different signs, you need to remember that $i^2 = -1$. For example, if you are multiplying $(3-5i)$ and $(5+7i)$, you need to use the formula:

$(3-5i)(5+7i) = (3 \cdot 5 - (-5) \cdot 7) + (3 \cdot 7 + (-5) \cdot 5)i$

Simplifying the expression, you get:

$(3-5i)(5+7i) = (15 + 35) + (21 - 25)i$

Combining like terms, you get:

$(3-5i)(5+7i) = 50 - 4i$

Q: Can I multiply complex numbers with the same sign?

A: Yes, you can multiply complex numbers with the same sign. For example, if you are multiplying $(3+5i)$ and $(5+7i)$, you can use the formula:

$(3+5i)(5+7i) = (3 \cdot 5 + 5 \cdot 7) + (3 \cdot 7 + 5 \cdot 5)i$

Simplifying the expression, you get:

$(3+5i)(5+7i) = (15 + 35) + (21 + 25)i$

Combining like terms, you get:

$(3+5i)(5+7i) = 50 + 46i$

Q: How do I multiply complex numbers with zero?

A: When multiplying a complex number with zero, the result is always zero. For example, if you are multiplying $(3-5i)$ and $0$, you get:

$(3-5i)(0) = 0$

Q: Can I multiply complex numbers with imaginary numbers only?

A: Yes, you can multiply complex numbers with imaginary numbers only. For example, if you are multiplying $(5i)$ and $(7i)$, you can use the formula:

$(5i)(7i) = 5 \cdot 7 \cdot i^2$

Since $i^2 = -1$, you get:

$(5i)(7i) = -35$

Conclusion

Multiplying complex numbers is a fundamental operation in complex number arithmetic. By using the distributive property of multiplication over addition, we can multiply complex numbers and simplify the resulting expression. In this article, we answered some frequently asked questions about multiplying complex numbers and provided examples to illustrate the concepts.

Additional Tips

Here are some additional tips for multiplying complex numbers:

• Always use the distributive property of multiplication over addition when multiplying complex numbers.
• Remember that $i^2 = -1$ when multiplying complex numbers with imaginary numbers.
• Simplify the expression by combining like terms.
• Check your work by multiplying the complex numbers in the opposite order.

We hope this article has provided a clear understanding of how to multiply complex numbers and has answered some of the most frequently asked questions about this topic.