Simplify The Given Expression Into The Form $a + B I$, Where $a$ And \$b$[/tex\] Are Rational Numbers.$2(-36 - 3i) + (5 + 2i)(12 - 2i)$

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Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. The given expression $2(-36 - 3i) + (5 + 2i)(12 - 2i)$ is a combination of two terms, each containing complex numbers. Our goal is to simplify this expression into the form $a + bi$, where $a$ and $b$ are rational numbers.

Understanding Complex Numbers

Before we dive into simplifying the given expression, let's quickly review the concept of complex numbers. A complex number is a number that can be expressed in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. The imaginary unit $i$ is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$. Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, but with some additional rules.

Distributive Property and FOIL Method

To simplify the given expression, we need to apply the distributive property and the FOIL method. The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. The FOIL method is a technique used to multiply two binomials, which stands for "First, Outer, Inner, Last". It states that to multiply two binomials, we multiply the first terms, then the outer terms, then the inner terms, and finally the last terms.

Simplifying the First Term

Let's start by simplifying the first term, $2(-36 - 3i)$. Using the distributive property, we can rewrite this as $-72 - 6i$.

Simplifying the Second Term

Now, let's simplify the second term, $(5 + 2i)(12 - 2i)$. We can use the FOIL method to multiply these two binomials. First, we multiply the first terms: $5 \cdot 12 = 60$. Then, we multiply the outer terms: $5 \cdot (-2i) = -10i$. Next, we multiply the inner terms: $2i \cdot 12 = 24i$. Finally, we multiply the last terms: $2i \cdot (-2i) = -4i^2$. Since $i^2 = -1$, we can replace $-4i^2$ with $4$. Therefore, the simplified form of the second term is $60 - 10i + 24i + 4$.

Combining Like Terms

Now that we have simplified both terms, we can combine like terms. The first term is $-72 - 6i$, and the second term is $60 - 10i + 24i + 4$. We can combine the real parts: $-72 + 60 + 4 = -8$. We can also combine the imaginary parts: $-6i - 10i + 24i = 8i$. Therefore, the simplified form of the given expression is $-8 + 8i$.

Conclusion

In this article, we simplified the given expression $2(-36 - 3i) + (5 + 2i)(12 - 2i)$ into the form $a + bi$, where $a$ and $b$ are rational numbers. We applied the distributive property and the FOIL method to simplify the expression, and then combined like terms to arrive at the final answer. This problem demonstrates the importance of simplifying complex expressions in mathematics, and how it can be achieved using various techniques and rules.

Final Answer

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

Related Topics

  • Simplifying complex expressions
  • Distributive property
  • FOIL method
  • Complex numbers
  • Rational numbers

Example Problems

  • Simplify the expression $3(2 - 5i) + (4 + 2i)(1 - 3i)$.
  • Simplify the expression $2(1 + 4i) + (3 - 2i)(2 + 5i)$.
  • Simplify the expression $4(3 - 2i) + (2 + 3i)(1 - 4i)$.

Practice Problems

  • Simplify the expression $5(2 - 3i) + (3 + 2i)(1 - 4i)$.
  • Simplify the expression $3(1 + 2i) + (2 - 3i)(4 + 5i)$.
  • Simplify the expression $2(3 - 4i) + (1 + 2i)(2 - 3i)$.

Solutions

  • Simplify the expression $3(2 - 5i) + (4 + 2i)(1 - 3i)$: $-9 - 15i + 4 - 8i + 2i - 6i^2$, which simplifies to $-1 - 17i$.
  • Simplify the expression $2(1 + 4i) + (3 - 2i)(2 + 5i)$: $2 + 8i + 6 + 10i - 4i - 10i^2$, which simplifies to $8 + 14i$.
  • Simplify the expression $4(3 - 2i) + (2 + 3i)(1 - 4i)$: $12 - 8i + 2 - 6i - 12i^2$, which simplifies to $14 - 16i$.

Tips and Tricks

  • When simplifying complex expressions, always start by applying the distributive property and the FOIL method.
  • Combine like terms carefully to avoid errors.
  • Use the imaginary unit $i$ to simplify expressions involving $i^2$.
  • Practice simplifying complex expressions regularly to become proficient in this skill.

Introduction

In our previous article, we simplified the given expression $2(-36 - 3i) + (5 + 2i)(12 - 2i)$ into the form $a + bi$, where $a$ and $b$ are rational numbers. In this article, we will answer some frequently asked questions related to simplifying complex expressions.

Q: What is the distributive property, and how is it used in simplifying complex expressions?

A: The distributive property is a mathematical rule that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property is used to simplify complex expressions by distributing the coefficients to the terms inside the parentheses.

Q: What is the FOIL method, and how is it used in simplifying complex expressions?

A: The FOIL method is a technique used to multiply two binomials. It stands for "First, Outer, Inner, Last", and it is used to multiply the first terms, then the outer terms, then the inner terms, and finally the last terms.

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

A: To simplify complex expressions involving imaginary numbers, you need to use the imaginary unit $i$, which is defined as the square root of $-1$. You can also use the fact that $i^2 = -1$ to simplify expressions involving $i^2$.

Q: What are some common mistakes to avoid when simplifying complex expressions?

A: Some common mistakes to avoid when simplifying complex expressions include:

  • Not distributing the coefficients correctly
  • Not combining like terms correctly
  • Not using the imaginary unit $i$ correctly
  • Not simplifying expressions involving $i^2$ correctly

Q: How can I practice simplifying complex expressions?

A: You can practice simplifying complex expressions by working on example problems and exercises. You can also try simplifying complex expressions on your own and then check your answers with a calculator or a reference solution.

Q: What are some real-world applications of simplifying complex expressions?

A: Simplifying complex expressions has many real-world applications, including:

  • Electrical engineering: Simplifying complex expressions is used to analyze and design electrical circuits.
  • Computer science: Simplifying complex expressions is used to optimize algorithms and data structures.
  • Physics: Simplifying complex expressions is used to solve problems in mechanics, electromagnetism, and quantum mechanics.

Q: Can I use a calculator to simplify complex expressions?

A: Yes, you can use a calculator to simplify complex expressions. However, it's always a good idea to check your answers with a reference solution to make sure you understand the steps involved in simplifying the expression.

Q: How can I simplify complex expressions involving fractions?

A: To simplify complex expressions involving fractions, you need to use the fact that fractions can be multiplied and divided just like real numbers. You can also use the fact that fractions can be simplified by canceling out common factors.

Q: What are some tips for simplifying complex expressions involving multiple variables?

A: Some tips for simplifying complex expressions involving multiple variables include:

  • Use the distributive property and the FOIL method to simplify expressions involving multiple variables.
  • Combine like terms carefully to avoid errors.
  • Use the imaginary unit $i$ to simplify expressions involving $i^2$.
  • Practice simplifying complex expressions involving multiple variables regularly to become proficient in this skill.

Conclusion

In this article, we answered some frequently asked questions related to simplifying complex expressions. We covered topics such as the distributive property, the FOIL method, and simplifying complex expressions involving imaginary numbers. We also provided some tips and tricks for simplifying complex expressions, including practicing with example problems and exercises.