Find The Following Product And Write The Result In Standard Form, \[$a + Bi\$\].\[$(-2+i)(6+i)\$\]\[$(-2+i)(6+i) =\$\] \[$\square\$\]

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Introduction

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Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will focus on multiplying complex numbers, which is a crucial operation in complex number arithmetic.

What are Complex Numbers?

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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 $i^2 = -1$. The real part of a complex number is denoted by $a$, and the imaginary part is denoted by $b$. For example, $3 + 4i$ is a complex number with real part $3$ and imaginary part $4$.

Multiplying Complex Numbers

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To multiply two complex numbers, we can 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:

$z_1 \cdot z_2 = (a + bi) \cdot (c + di)$

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

$z_1 \cdot z_2 = ac + adi + bci + bdi^2$

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

$z_1 \cdot z_2 = ac + adi + bci - bd$

Combining like terms, we get:

$z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$

Example: Multiplying Two Complex Numbers

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Let's consider the product of two complex numbers $(-2 + i)$ and $(6 + i)$. Using the formula for multiplying complex numbers, we get:

$(-2 + i) \cdot (6 + i) = (-2 \cdot 6 - 1 \cdot 1) + (-2 \cdot 1 + 1 \cdot 6)i$

Simplifying the expression, we get:

$(-2 + i) \cdot (6 + i) = (-12 - 1) + (-2 + 6)i$

Combining like terms, we get:

$(-2 + i) \cdot (6 + i) = -13 + 4i$

Conclusion

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In this article, we have discussed the concept of complex numbers and the operation of multiplying complex numbers. We have also provided an example of multiplying two complex numbers, $(-2 + i)$ and $(6 + i)$. The result of the multiplication is $-13 + 4i$, which is a complex number with real part $-13$ and imaginary part $4$.

Final Answer

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The final answer is $\boxed{-13 + 4i}$.

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Introduction

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Multiplying complex numbers is a fundamental operation in complex number arithmetic. In our previous article, we discussed the concept of complex numbers and the operation of multiplying complex numbers. In this article, we will answer some frequently asked questions about complex number multiplication.

Q: What is the formula for multiplying complex numbers?

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A: The formula for multiplying complex numbers is:

$z_1 \cdot z_2 = (a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i$

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

Q: How do I multiply two complex numbers?

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A: To multiply two complex numbers, you can use the distributive property of multiplication over addition. Multiply each term in the first complex number by each term in the second complex number, and then combine like terms.

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

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A: The main difference between multiplying complex numbers and multiplying real numbers is that complex numbers have an imaginary part, which is denoted by $i$. When multiplying complex numbers, you need to take into account the imaginary part and use the formula for multiplying complex numbers.

Q: Can I multiply a complex number by a real number?

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A: Yes, you can multiply a complex number by a real number. In this case, the real number can be treated as a complex number with an imaginary part of $0$. For example, multiplying a complex number $a + bi$ by a real number $c$ is equivalent to multiplying $a + bi$ by $c + 0i$.

Q: How do I simplify the product of two complex numbers?

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A: To simplify the product of two complex numbers, you can use the formula for multiplying complex numbers and then combine like terms. You can also use the fact that $i^2 = -1$ to simplify the expression.

Q: What is the result of multiplying two complex numbers with the same real part?

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A: If two complex numbers have the same real part, the result of multiplying them will also have the same real part. The imaginary part of the result will be the sum of the imaginary parts of the two complex numbers.

Q: Can I multiply two complex numbers with different magnitudes?

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A: Yes, you can multiply two complex numbers with different magnitudes. The magnitude of the result will be the product of the magnitudes of the two complex numbers.

Q: How do I multiply a complex number by a complex number with a different magnitude?

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A: To multiply a complex number by a complex number with a different magnitude, you can use the formula for multiplying complex numbers and then simplify the expression.

Q: What is the result of multiplying a complex number by its conjugate?

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A: If a complex number is multiplied by its conjugate, the result will be a real number. The conjugate of a complex number $a + bi$ is $a - bi$.

Q: Can I multiply a complex number by its reciprocal?

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A: Yes, you can multiply a complex number by its reciprocal. The reciprocal of a complex number $a + bi$ is $\frac{1}{a + bi}$.

Q: How do I multiply a complex number by its reciprocal?

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A: To multiply a complex number by its reciprocal, you can use the formula for multiplying complex numbers and then simplify the expression.

Q: What is the result of multiplying a complex number by its reciprocal?

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A: If a complex number is multiplied by its reciprocal, the result will be $1$.