Perform Operations And Express Result In A + Bi Form

Understanding Complex Numbers

Before we delve into solving the specific problem, let's first establish a solid understanding of complex numbers. 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, defined as the square root of -1 (i = √-1). The real part of the complex number is a, and the imaginary part is b. Complex numbers extend the real number system by including the imaginary unit, which allows us to work with the square roots of negative numbers. This is crucial in various fields, including mathematics, physics, and engineering, where complex numbers arise naturally in many problems.

The significance of complex numbers lies in their ability to provide solutions to equations that have no real solutions. For instance, the equation x² + 1 = 0 has no real solutions because the square of any real number is non-negative. However, introducing the imaginary unit i allows us to express the solutions as x = ±i. This capability expands the scope of mathematical problem-solving, enabling us to address scenarios that were previously unsolvable within the realm of real numbers. Furthermore, complex numbers possess a rich algebraic structure, meaning that they can be added, subtracted, multiplied, and divided, following specific rules that ensure consistent and meaningful results. These rules are essential for performing operations with complex numbers and obtaining correct answers.

When performing operations with complex numbers, it's essential to remember that i² = -1. This identity is fundamental and is used extensively in simplifying expressions involving complex numbers. For example, when multiplying two complex numbers, we often encounter terms with i², which can then be replaced with -1, leading to a simplified result. Additionally, understanding the properties of complex conjugates is vital for division. The complex conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate eliminates the imaginary part in the denominator, making it easier to express the result in the standard form a + bi. These basic principles form the foundation for more complex calculations and manipulations involving complex numbers.

Problem Statement and Initial Simplification

Our primary task is to perform the indicated operations and write the result in the form a + bi. The expression we are given is: $(\sqrt{2} - \sqrt{-2})(\sqrt{32} + \sqrt{-2})$. To approach this problem effectively, we first need to simplify the square roots of negative numbers. Remember, the square root of a negative number can be expressed using the imaginary unit i, where i = √-1. Therefore, we can rewrite √-2 as √(2 * -1) = √2 * √-1 = √2 * i. This transformation is a crucial step in dealing with complex numbers, as it allows us to separate the real and imaginary parts and proceed with algebraic manipulations more easily.

Now, let's rewrite the original expression using this simplification. We have: (√2 - √2 * i)(√32 + √2 * i). Next, we can simplify √32. Since 32 = 16 * 2, we have √32 = √(16 * 2) = √16 * √2 = 4√2. Substituting this back into our expression, we get: (√2 - √2 * i)(4√2 + √2 * i). This simplified form is much easier to work with, as it contains only real numbers and the imaginary unit i. The simplification process has transformed the original expression into a more manageable form, setting the stage for the next steps in the calculation.

The significance of simplifying square roots of negative numbers cannot be overstated. This process is essential for correctly handling complex numbers and performing operations on them. By expressing the square roots of negative numbers in terms of i, we ensure that we are working within the framework of complex number algebra, which has specific rules and properties that must be followed. Failure to perform this simplification can lead to incorrect results and a misunderstanding of the underlying mathematical principles. Therefore, it's crucial to always simplify square roots of negative numbers before attempting to perform any other operations with complex numbers.

Expanding the Expression

With the expression now simplified to (√2 - √2 * i)(4√2 + √2 * i), we proceed by expanding it using the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Let's break down the expansion step by step:

1. First: Multiply the first terms in each binomial: √2 * 4√2 = 4 * (√2 * √2) = 4 * 2 = 8.
2. Outer: Multiply the outer terms: √2 * √2 * i = 2i.
3. Inner: Multiply the inner terms: -√2 * i * 4√2 = -4 * (√2 * √2) * i = -4 * 2 * i = -8i.
4. Last: Multiply the last terms: -√2 * i * √2 * i = -2 * i².

Combining these results, we get: 8 + 2i - 8i - 2i². This expanded form provides us with a clearer view of the terms that need further simplification. The distributive property is a fundamental algebraic technique that allows us to multiply polynomials, and its application here is crucial for correctly expanding the expression involving complex numbers. Each term resulting from the multiplication contributes to the final result, and it's important to keep track of the signs and coefficients to avoid errors.

The expansion process is a critical step in simplifying expressions involving multiple terms. It transforms the product of binomials into a sum of individual terms, making it easier to combine like terms and simplify the expression further. In the context of complex numbers, the expansion is particularly important because it often leads to terms involving i², which can then be replaced with -1, a key step in expressing the result in the standard form a + bi. The careful application of the distributive property ensures that the expansion is performed correctly and that all terms are accounted for.

Simplifying and Combining Terms

After expanding the expression, we have 8 + 2i - 8i - 2i². The next step is to simplify this by dealing with the i² term and combining like terms. Remember that i² = -1, so we can replace -2i² with -2 * (-1) = 2. This substitution is a fundamental aspect of working with complex numbers, as it allows us to eliminate the imaginary unit raised to a power and express the result in terms of real numbers and i.

Substituting i² = -1 into our expression, we get: 8 + 2i - 8i + 2. Now, we can combine the real parts (8 and 2) and the imaginary parts (2i and -8i) separately. Combining the real parts gives us 8 + 2 = 10. Combining the imaginary parts gives us 2i - 8i = -6i. Therefore, the simplified expression is 10 - 6i. This form is now in the standard form a + bi, where a is the real part (10) and b is the imaginary part (-6).

The process of combining like terms is a fundamental algebraic technique that is essential for simplifying expressions. It involves identifying terms that have the same variable parts and adding or subtracting their coefficients. In the context of complex numbers, this means combining the real parts and the imaginary parts separately. This process is crucial for expressing the result in the standard form a + bi, which is the preferred way to represent complex numbers. By combining like terms, we reduce the complexity of the expression and make it easier to interpret and use.

Final Result in a + bi Form

After performing the indicated operations and simplifying the expression, we have arrived at the result: 10 - 6i. This result is in the standard form a + bi, where a = 10 and b = -6. The real part of the complex number is 10, and the imaginary part is -6. This final form is clear, concise, and easily interpretable, which is why it is the standard way to represent complex numbers. Expressing complex numbers in this form allows for easy comparison and further manipulation.

The importance of expressing the final result in the standard form a + bi cannot be overstated. This form provides a clear separation between the real and imaginary parts of the complex number, making it easy to identify and work with each part independently. It also facilitates further calculations and comparisons, as the real and imaginary parts can be treated as separate components. The standard form is universally recognized and used in mathematics, physics, engineering, and other fields, making it the preferred way to represent complex numbers.

In conclusion, we have successfully performed the indicated operations on the given expression and written the result in the form a + bi. The steps involved simplification of square roots of negative numbers, expansion using the distributive property, simplification by substituting i² = -1, and combining like terms. The final result, 10 - 6i, is a complex number with a real part of 10 and an imaginary part of -6. This process demonstrates the fundamental techniques for working with complex numbers and highlights the importance of following the correct algebraic procedures to arrive at the correct answer.

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Perform Operations and Express Result in a + bi Form