Equivalent Expressions Simplifying Complex Numbers Step By Step

In the realm of mathematics, navigating complex expressions can often feel like traversing a labyrinth. The key to success lies in a systematic approach, coupled with a solid understanding of fundamental principles. This article serves as a comprehensive guide to simplifying complex expressions, focusing on the expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. We will dissect the expression step-by-step, revealing the underlying concepts and demonstrating how to arrive at the equivalent forms. By the end of this exploration, you will be equipped with the tools and knowledge to confidently tackle similar challenges.

Demystifying the Expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$

Our journey begins with the expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. At first glance, this may appear daunting, but by breaking it down into manageable components, we can unravel its complexity. The expression involves square roots of both positive and negative numbers, which introduces the concept of imaginary numbers. Our primary goal is to simplify this expression by evaluating the square roots, combining like terms, and ultimately arriving at its equivalent forms. This process involves a meticulous application of mathematical rules and properties, ensuring that each step is logically sound and contributes to the overall simplification.

Step 1: Evaluating Square Roots

The cornerstone of simplifying this expression is evaluating the square roots. Let's begin with the positive square roots. The square root of 9, denoted as $\sqrt{9}$, is 3, since 3 multiplied by itself equals 9. Similarly, the square root of 576, $\sqrt{576}$, is 24, as 24 times 24 is 576. Now, let's turn our attention to the square roots of negative numbers. This is where the concept of imaginary numbers comes into play. The square root of -1 is defined as the imaginary unit, denoted by the letter 'i'. This definition is crucial for handling square roots of negative numbers. The square root of -4, $\sqrt{-4}$, can be expressed as $\sqrt{4 * -1}$, which simplifies to $\sqrt{4} * \sqrt{-1}$, or 2i. Likewise, the square root of -64, $\sqrt{-64}$, can be written as $\sqrt{64 * -1}$, which simplifies to $\sqrt{64} * \sqrt{-1}$, or 8i. With these square roots evaluated, we can substitute them back into the original expression.

Step 2: Substituting Evaluated Square Roots

Having evaluated the square roots, our next step is to substitute these values back into the original expression. The expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$ now transforms into (-3 + 2i) - (2 * 24 + 8i). This substitution is a pivotal step as it replaces the square root terms with their numerical and imaginary equivalents, paving the way for further simplification. It's essential to ensure accurate substitution to maintain the integrity of the expression and avoid errors in subsequent calculations. The substitution process bridges the gap between the initial complex form and a more manageable algebraic form, making it easier to apply the rules of arithmetic and algebra.

Step 3: Simplifying the Expression

With the square roots substituted, we can now focus on simplifying the expression. This involves performing the arithmetic operations and combining like terms. First, we address the multiplication within the parentheses: 2 * 24 equals 48. So, the expression becomes (-3 + 2i) - (48 + 8i). Next, we distribute the negative sign in front of the second parenthesis, which changes the signs of the terms inside: -3 + 2i - 48 - 8i. Now, we combine the real terms (-3 and -48) and the imaginary terms (2i and -8i). Combining the real terms, -3 - 48 equals -51. Combining the imaginary terms, 2i - 8i equals -6i. Thus, the simplified expression is -51 - 6i. This process highlights the importance of adhering to the order of operations and accurately applying the distributive property. The simplification process transforms the expression into its most concise form, making it easier to interpret and compare with other expressions.

Step 4: Identifying Equivalent Expressions

Having simplified the original expression to -51 - 6i, we can now identify the equivalent expressions from the given options. This involves comparing the simplified form with each option and determining which ones match. The key is to recognize that equivalent expressions represent the same mathematical value, even if they appear different on the surface. To verify equivalence, we may need to perform additional algebraic manipulations on the options to see if they can be transformed into the simplified form -51 - 6i. This step reinforces the concept of mathematical equivalence and the ability to recognize different representations of the same quantity. By identifying equivalent expressions, we demonstrate a deep understanding of the underlying mathematical principles and the flexibility to work with expressions in various forms.

Analyzing the Given Options for Equivalence

Now, let's meticulously examine each of the provided options to determine which ones are equivalent to our simplified expression, -51 - 6i.

• Option 1: -3 - 2i - 2(24) + 8i

  We can simplify this expression by performing the multiplication: -3 - 2i - 48 + 8i. Combining the real terms (-3 and -48) gives -51. Combining the imaginary terms (-2i and 8i) gives +6i. Thus, the simplified form of this option is -51 + 6i. Comparing this with our simplified expression, -51 - 6i, we see that they are not equivalent. The imaginary parts have opposite signs.

• Option 2: -51 - 6i

  This option directly matches our simplified expression, -51 - 6i. Therefore, this option is equivalent.

• Option 3: -3 + 2i + 2(24) + 8i

  Simplifying this expression, we first perform the multiplication: -3 + 2i + 48 + 8i. Combining the real terms (-3 and 48) gives 45. Combining the imaginary terms (2i and 8i) gives 10i. Thus, the simplified form of this option is 45 + 10i. This is clearly not equivalent to -51 - 6i.

• Option 4: -3 + 2i - 2(24) - 8i

  Simplifying this expression, we perform the multiplication: -3 + 2i - 48 - 8i. Combining the real terms (-3 and -48) gives -51. Combining the imaginary terms (2i and -8i) gives -6i. Thus, the simplified form of this option is -51 - 6i. This matches our simplified expression, so this option is equivalent.

• Option 5: -51 + 6i

  As we determined in the analysis of Option 1, this expression simplifies to -51 + 6i, which is not equivalent to -51 - 6i. The signs of the imaginary parts differ.

• Option 6: 45 + 10i

  This expression, as we saw in the analysis of Option 3, is 45 + 10i, which is not equivalent to -51 - 6i. The real and imaginary parts are different.

Conclusion: Equivalent Expressions Identified

Through a meticulous step-by-step simplification process and a careful comparison of the given options, we have successfully identified the expressions that are equivalent to $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. The equivalent expressions are:

• -51 - 6i
• -3 + 2i - 2(24) - 8i

This exercise underscores the importance of understanding the properties of complex numbers, the order of operations, and the concept of mathematical equivalence. By mastering these principles, one can confidently navigate the complexities of algebraic expressions and arrive at accurate solutions.

Key Takeaways and Further Exploration

This exploration into simplifying complex expressions has highlighted several key takeaways:

• Understanding Complex Numbers: The ability to work with imaginary numbers (involving the imaginary unit 'i') is crucial for simplifying expressions containing square roots of negative numbers.
• Order of Operations: Adhering to the order of operations (PEMDAS/BODMAS) is essential for accurate simplification.
• Combining Like Terms: Simplifying expressions often involves combining real and imaginary terms separately.
• Recognizing Equivalence: Understanding that different expressions can represent the same mathematical value is key to identifying equivalent forms.

To further enhance your understanding, consider exploring the following topics:

• Operations with Complex Numbers: Delve deeper into addition, subtraction, multiplication, and division of complex numbers.
• The Complex Plane: Visualize complex numbers graphically on the complex plane.
• Euler's Formula: Explore the fascinating connection between complex numbers and trigonometric functions through Euler's formula.

By continuing your exploration of these concepts, you will build a strong foundation in complex number theory and enhance your mathematical problem-solving skills.

Which of the following expressions are equivalent to $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$? Select all that apply.

Equivalent Expressions Simplify Complex Numbers Step-by-Step