Simplify The Expression: \[$(6-2i)(4-5i)\$\].A. \[$34-17i\$\]B. \[$14-17i\$\]C. \[$34-38i\$\]D. \[$14-38i\$\]

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Introduction

In this article, we will simplify the given expression $(6-2i)(4-5i)$ using the distributive property of multiplication over addition. This involves multiplying each term in the first expression by each term in the second expression and then combining like terms.

Step 1: Multiply Each Term

To simplify the expression, we need to multiply each term in the first expression $(6-2i)$ by each term in the second expression $(4-5i)$. This will result in four terms: $6 \cdot 4$, $6 \cdot (-5i)$, $(-2i) \cdot 4$, and $(-2i) \cdot (-5i)$.

Step 2: Apply the Distributive Property

Using the distributive property, we can rewrite the expression as:

$(6-2i)(4-5i) = 6 \cdot 4 + 6 \cdot (-5i) + (-2i) \cdot 4 + (-2i) \cdot (-5i)$

Step 3: Simplify Each Term

Now, we can simplify each term:

$6 \cdot 4 = 24$

$6 \cdot (-5i) = -30i$

$(-2i) \cdot 4 = -8i$

$(-2i) \cdot (-5i) = 10i^2$

Step 4: Combine Like Terms

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

$10i^2 = 10(-1) = -10$

Now, we can combine like terms:

$24 - 30i - 8i - 10$

Step 5: Simplify the Expression

Combining like terms, we get:

$24 - 38i - 10$

Simplifying further, we get:

$14 - 38i$

Conclusion

Therefore, the simplified expression is $(14 - 38i)$.

Answer

The correct answer is:

• D. ${14-38i\$}

Explanation

The correct answer is obtained by multiplying each term in the first expression by each term in the second expression and then combining like terms. The final expression is $(14 - 38i)$, which matches option D.

Tips and Tricks

When simplifying complex expressions, it's essential 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 multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following these steps and using the distributive property, you can simplify complex expressions like $(6-2i)(4-5i)$.

Practice Problems

Try simplifying the following expressions:

1. $(3+4i)(2-3i)$
2. $(5-2i)(3+4i)$
3. $(2+3i)(4-2i)$

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Introduction

In our previous article, we simplified the expression $(6-2i)(4-5i)$ using the distributive property of multiplication over addition. In this article, we will answer some frequently asked questions (FAQs) related to simplifying complex expressions.

Q: What is the distributive property of multiplication over addition?

A: The distributive property of multiplication over addition states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

This property allows us to multiply each term in the first expression by each term in the second expression.

Q: How do I apply the distributive property to simplify complex expressions?

A: To apply the distributive property, follow these steps:

1. Multiply each term in the first expression by each term in the second expression.
2. Combine like terms by adding or subtracting the coefficients of the same variable.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I simplify complex expressions with imaginary numbers?

A: To simplify complex expressions with imaginary numbers, follow these steps:

1. Multiply each term in the first expression by each term in the second expression.
2. Combine like terms by adding or subtracting the coefficients of the same variable.
3. Simplify any expressions involving $i^2$ by substituting $i^2 = -1$.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed without any imaginary part, such as 3 or -4. An imaginary number is a number that can be expressed with an imaginary part, such as $2i$ or $-3i$.

Q: How do I simplify expressions with complex conjugates?

A: Complex conjugates are expressions that have the same real part but opposite imaginary parts. For example, $2+3i$ and $2-3i$ are complex conjugates. To simplify expressions with complex conjugates, follow these steps:

1. Multiply the complex conjugates together.
2. Simplify the resulting expression.

Q: What is the significance of 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, which includes all numbers of the form $a+bi$, where $a$ and $b$ are real numbers.

Conclusion

Simplifying complex expressions requires a thorough understanding of the distributive property, like terms, and imaginary numbers. By following the steps outlined in this article, you can simplify complex expressions and solve problems involving complex numbers.

Practice Problems

Try simplifying the following expressions:

1. $(3+4i)(2-3i)$
2. $(5-2i)(3+4i)$
3. $(2+3i)(4-2i)$

Use the distributive property and combine like terms to simplify each expression.

Additional Resources

For more information on simplifying complex expressions, check out the following resources:

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

By practicing and mastering the skills outlined in this article, you can become proficient in simplifying complex expressions and solving problems involving complex numbers.