Simplify The Expression:$\[(3 + 2i)(4 - 5i)\\]

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Introduction

In mathematics, the process of simplifying complex expressions is a crucial skill that is often required in various fields, including algebra, calculus, and engineering. One of the most common methods of simplifying complex expressions is by using the distributive property, which allows us to multiply each term in one expression by each term in another expression. In this article, we will focus on simplifying the expression (3 + 2i)(4 - 5i) using the distributive property.

Understanding Complex Numbers

Before we can simplify the expression, it is essential to understand what complex numbers are. 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. In the given expression, 3 + 2i is a complex number, where 3 is the real part and 2i is the imaginary part.

Applying the Distributive Property

To simplify the expression (3 + 2i)(4 - 5i), we will use the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this case, we can rewrite the expression as:

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

Multiplying Each Term

Now, we will multiply each term in the first expression by each term in the second expression:

3(4 - 5i) = 12 - 15i 2i(4 - 5i) = 8i - 10i^2

Simplifying the Expression

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

8i - 10i^2 = 8i - 10(-1) = 8i + 10

Combining Like Terms

Now, we can combine the like terms in the expression:

12 - 15i + 8i + 10 = 22 - 7i

Conclusion

In this article, we have simplified the expression (3 + 2i)(4 - 5i) using the distributive property. We have shown that the expression can be simplified to 22 - 7i by multiplying each term in the first expression by each term in the second expression and combining like terms. This process is essential in mathematics and is used in various fields, including algebra, calculus, and engineering.

Tips and Tricks

  • When simplifying complex expressions, it is essential to use the distributive property to multiply each term in one expression by each term in another expression.
  • When combining like terms, make sure to combine the real parts and the imaginary parts separately.
  • When working with complex numbers, it is essential to remember that i^2 = -1.

Real-World Applications

The process of simplifying complex expressions is used in various fields, including:

  • Algebra: Simplifying complex expressions is a crucial skill in algebra, where it is used to solve equations and inequalities.
  • Calculus: Simplifying complex expressions is used in calculus to find derivatives and integrals.
  • Engineering: Simplifying complex expressions is used in engineering to design and analyze complex systems.

Final Thoughts

In conclusion, simplifying complex expressions is a crucial skill that is used in various fields, including algebra, calculus, and engineering. By using the distributive property and combining like terms, we can simplify complex expressions and solve problems in mathematics and other fields.

Introduction

In our previous article, we simplified the expression (3 + 2i)(4 - 5i) using the distributive property. In this article, we will answer some frequently asked questions related to simplifying complex expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to multiply each term in one expression by each term in another expression. It is used to simplify complex expressions and is a crucial skill in mathematics.

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

A: To apply the distributive property, you need to multiply each term in one expression by each term in another expression. For example, if you have the expression (a + b)(c + d), you would multiply a by c, a by d, b by c, and b by d.

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

A: The real part of a complex number is the part that is not multiplied by i, while the imaginary part is the part that is multiplied by i. For example, in the complex number 3 + 2i, 3 is the real part and 2i is the imaginary part.

Q: How do I combine like terms in a complex expression?

A: To combine like terms, you need to add or subtract the real parts and the imaginary parts separately. For example, if you have the expression 2 + 3i + 4 + 5i, you would combine the real parts (2 + 4) and the imaginary parts (3i + 5i).

Q: What is the significance of i^2 = -1 in complex numbers?

A: i^2 = -1 is a fundamental property of complex numbers that allows us to simplify complex expressions. It is used to eliminate the square of i and replace it with -1.

Q: Can I use the distributive property to simplify expressions with more than two terms?

A: Yes, you can use the distributive property to simplify expressions with more than two terms. For example, if you have the expression (a + b + c)(d + e + f), you would multiply each term in the first expression by each term in the second expression.

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

A: To simplify complex expressions with negative numbers, you need to follow the same steps as simplifying complex expressions with positive numbers. For example, if you have the expression (-3 + 2i)(4 - 5i), you would multiply each term in the first expression by each term in the second expression.

Tips and Tricks

  • When simplifying complex expressions, make sure to use the distributive property to multiply each term in one expression by each term in another expression.
  • When combining like terms, make sure to combine the real parts and the imaginary parts separately.
  • When working with complex numbers, remember that i^2 = -1.

Real-World Applications

The process of simplifying complex expressions is used in various fields, including:

  • Algebra: Simplifying complex expressions is a crucial skill in algebra, where it is used to solve equations and inequalities.
  • Calculus: Simplifying complex expressions is used in calculus to find derivatives and integrals.
  • Engineering: Simplifying complex expressions is used in engineering to design and analyze complex systems.

Final Thoughts

In conclusion, simplifying complex expressions is a crucial skill that is used in various fields, including algebra, calculus, and engineering. By using the distributive property and combining like terms, we can simplify complex expressions and solve problems in mathematics and other fields.