Simplify The Expression Below As Much As Possible:$\[(6 - 4i) + (-3 + 4i) - (2 - 8i)\\]A. \[$1 + 8i\$\] B. \[$5 - 8i\$\] C. \[$5 + 8i\$\] D. \[$1 - 8i\$\]

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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 ${(6 - 4i) + (-3 + 4i) - (2 - 8i)}$.

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, the complex number $3 + 4i$ has a real part of $3$ and an imaginary part of $4$.

Simplifying Complex Numbers

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Simplifying complex numbers involves combining like terms and performing arithmetic operations on the real and imaginary parts. To simplify the expression ${(6 - 4i) + (-3 + 4i) - (2 - 8i)}$, we need to follow the order of operations (PEMDAS):

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.

Step-by-Step Solution

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Step 1: Combine Like Terms

The given expression can be rewritten as:

$${(6 - 4i) + (-3 + 4i) - (2 - 8i)}$$

$= (6 - 4i) + (-3 + 4i) - (2 - 8i)$

$= (6 - 4i) + (-3 + 4i) + (-2 + 8i)$

Now, we can combine like terms:

$= (6 - 3 - 2) + (-4i + 4i + 8i)$

$= 1 + 8i$

Step 2: Check the Answer Choices

We have simplified the expression to $1 + 8i$. Now, let's check the answer choices:

• A. $1 + 8i$
• B. $5 - 8i$
• C. $5 + 8i$
• D. $1 - 8i$

Only answer choice A matches our simplified expression.

Conclusion

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In this article, we simplified the complex number expression ${(6 - 4i) + (-3 + 4i) - (2 - 8i)}$ using the order of operations and combining like terms. We arrived at the simplified expression $1 + 8i$, which matches answer choice A.

Frequently Asked Questions

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Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is a mathematical constant that satisfies the equation $i^2 = -1$. It is used to extend the real number system to the complex number system.

Q: How do I simplify complex numbers?

A: To simplify complex numbers, follow the order of operations (PEMDAS): evaluate expressions inside parentheses, exponents, multiplication and division, and finally addition and subtraction.

Q: What is the difference between real and imaginary parts of a complex number?

A: The real part of a complex number is the part without the imaginary unit $i$, while the imaginary part is the part with the imaginary unit $i$.

Final Answer

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The final answer is $\boxed{1 + 8i}$.

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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 address some of the most frequently asked questions about complex numbers.

Q: What is the definition of a complex number?

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A: 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 $i^2 = -1$.

Q: What is the difference between real and imaginary parts of a complex number?

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A: The real part of a complex number is the part without the imaginary unit $i$, while the imaginary part is the part with the imaginary unit $i$. For example, in the complex number $3 + 4i$, the real part is $3$ and the imaginary part is $4$.

Q: How do I add complex numbers?

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A: To add complex numbers, simply add the real parts and the imaginary parts separately. For example, to add $2 + 3i$ and $4 + 5i$, we get:

$(2 + 3i) + (4 + 5i) = (2 + 4) + (3i + 5i) = 6 + 8i$

Q: How do I subtract complex numbers?

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A: To subtract complex numbers, simply subtract the real parts and the imaginary parts separately. For example, to subtract $4 + 5i$ from $2 + 3i$, we get:

$(2 + 3i) - (4 + 5i) = (2 - 4) + (3i - 5i) = -2 - 2i$

Q: How do I multiply complex numbers?

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A: To multiply complex numbers, we use the distributive property and the fact that $i^2 = -1$. For example, to multiply $2 + 3i$ and $4 + 5i$, we get:

$(2 + 3i)(4 + 5i) = 2(4 + 5i) + 3i(4 + 5i)$ $= 8 + 10i + 12i + 15i^2$ $= 8 + 22i - 15$ $= -7 + 22i$

Q: How do I divide complex numbers?

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A: To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. For example, to divide $2 + 3i$ by $4 + 5i$, we get:

$\frac{2 + 3i}{4 + 5i} = \frac{(2 + 3i)(4 - 5i)}{(4 + 5i)(4 - 5i)}$ $= \frac{8 - 10i + 12i - 15i^2}{16 - 25i^2}$ $= \frac{8 + 2i + 15}{16 + 25}$ $= \frac{23 + 2i}{41}$

Q: What is the conjugate of a complex number?

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A: The conjugate of a complex number $a + bi$ is $a - bi$. For example, the conjugate of $3 + 4i$ is $3 - 4i$.

Q: What is the modulus of a complex number?

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A: The modulus of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$. For example, the modulus of $3 + 4i$ is $\sqrt{3^2 + 4^2} = 5$.

Q: What is the argument of a complex number?

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A: The argument of a complex number $a + bi$ is the angle $\theta$ between the positive real axis and the line segment joining the origin to the point $(a, b)$. For example, the argument of $3 + 4i$ is $\tan^{-1}\left(\frac{4}{3}\right)$.

Q: What are some common applications of complex numbers?

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A: Complex numbers have numerous applications in various fields, including:

• Algebra: Complex numbers are used to solve polynomial equations and to study the properties of algebraic structures.
• Geometry: Complex numbers are used to study the properties of geometric shapes and to perform geometric transformations.
• Calculus: Complex numbers are used to study the properties of functions and to perform integration and differentiation.
• Physics: Complex numbers are used to study the properties of waves and to perform calculations in quantum mechanics and electromagnetism.

Conclusion

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In this article, we have addressed some of the most frequently asked questions about complex numbers. We have covered topics such as the definition of a complex number, the difference between real and imaginary parts, addition, subtraction, multiplication, and division of complex numbers, the conjugate and modulus of a complex number, and some common applications of complex numbers. We hope that this article has been helpful in clarifying any doubts you may have had about complex numbers.