Simplify (-4 - √-50) / 60 Express In A + Bi Form

In this article, we will delve into the process of performing indicated operations, specifically focusing on simplifying expressions involving complex numbers. Our primary example will be the expression:

$\frac{-4-\sqrt{-50}}{60}=$

This requires a solid understanding of how to manipulate square roots of negative numbers and express the result in the standard form of a complex number, which is $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $\sqrt{-1}$. Let's break down the steps involved in simplifying this expression.

Understanding Complex Numbers

Before we tackle the expression, let's lay the groundwork by understanding what complex numbers are. Complex numbers extend the real number system by including the imaginary unit i. The imaginary unit i is defined as the square root of -1, i.e., $i = \sqrt{-1}$. This definition allows us to work with the square roots of negative numbers, which are not defined in the real number system. A complex number is generally expressed in the form $a + bi$, where a is the real part and bi is the imaginary part. For example, in the complex number $3 + 4i$, 3 is the real part, and 4i is the imaginary part. Understanding this foundational concept is crucial for simplifying expressions involving square roots of negative numbers. The beauty of complex numbers lies in their ability to provide solutions to equations that have no real solutions. For instance, the equation $x^2 + 1 = 0$ has no real solutions because the square of any real number is non-negative. However, using complex numbers, we can find solutions such as $x = i$ and $x = -i$. This expansion of the number system opens up a whole new realm of mathematical possibilities and applications, especially in fields like electrical engineering, quantum mechanics, and fluid dynamics. When working with complex numbers, operations such as addition, subtraction, multiplication, and division are defined in a way that preserves the properties of real numbers while also accounting for the unique nature of the imaginary unit. For example, when adding or subtracting complex numbers, we simply add or subtract the real and imaginary parts separately. When multiplying complex numbers, we use the distributive property and the fact that $i^2 = -1$ to simplify the result. This allows us to manipulate and simplify complex expressions, ultimately leading to solutions that would be unattainable within the confines of the real number system alone. Thus, a thorough grasp of complex numbers is not only essential for solving specific mathematical problems but also for understanding broader concepts in various scientific and engineering disciplines.

Step-by-Step Simplification

Now, let’s simplify the expression $\frac{-4-\sqrt{-50}}{60}$ step by step. This process involves simplifying the square root of the negative number, factoring out common factors, and expressing the final answer in the standard complex number form $a + bi$. By carefully following each step, we can transform the initial expression into its simplest form, making it easier to work with and interpret. Each step builds upon the previous one, ensuring that we maintain mathematical accuracy and arrive at the correct solution. The goal is not just to find the answer but also to understand the underlying principles that govern the manipulation of complex numbers. This understanding will be invaluable when tackling more complex problems in the future. Therefore, let's proceed with the simplification process, paying close attention to the details and the reasoning behind each operation. This will not only help us solve the current problem but also enhance our overall mathematical proficiency.

1. Simplify the Square Root

The first step is to simplify the square root of -50. We can rewrite $\sqrt{-50}$ as $\sqrt{50 \cdot -1}$. Since $\sqrt{-1}$ is defined as i, we have $\sqrt{-50} = \sqrt{50} \cdot \sqrt{-1} = \sqrt{50}i$. Now, we simplify $\sqrt{50}$. We can factor 50 as $25 \cdot 2$, so $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$. Therefore, $\sqrt{-50} = 5\sqrt{2}i$. This step is crucial because it transforms the expression from one involving the square root of a negative number to one involving the imaginary unit i. By breaking down the square root into its constituent factors, we can isolate the imaginary component and express it in a more manageable form. This transformation is a fundamental technique in working with complex numbers, allowing us to apply the rules of algebra and arithmetic in a consistent manner. Furthermore, it highlights the importance of prime factorization and understanding the properties of square roots. By simplifying the square root, we pave the way for further simplification of the entire expression, making it easier to combine terms and express the final result in the standard complex number form. This step not only simplifies the expression but also deepens our understanding of the underlying mathematical principles at play.

2. Substitute Back into the Expression

Substitute $5\sqrt{2}i$ back into the original expression: $\frac{-4 - 5\sqrt{2}i}{60}$. This substitution is a critical step in the simplification process. By replacing the original square root term with its simplified form, we make the expression more manageable and easier to work with. This step ensures that we are dealing with the imaginary unit i explicitly, which is essential for expressing the final answer in the standard complex number form a + bi. Moreover, it allows us to see the structure of the expression more clearly, making it easier to identify common factors and apply further simplification techniques. The act of substituting back into the expression is a fundamental algebraic technique that is used in various mathematical contexts. It helps to maintain the equality of the expression while transforming it into a more usable form. In this case, the substitution allows us to proceed with the next step, which involves separating the real and imaginary parts and simplifying the fractions. Thus, this step is not just a mechanical replacement but a strategic move that brings us closer to the final simplified form of the expression.

3. Separate Real and Imaginary Parts

Next, we separate the real and imaginary parts of the fraction: $\frac{-4 - 5\sqrt{2}i}{60} = \frac{-4}{60} - \frac{5\sqrt{2}i}{60}$. Separating the real and imaginary parts is a fundamental technique when working with complex numbers. This step allows us to treat the real and imaginary components independently, making it easier to simplify each part. By dividing each term in the numerator by the denominator, we break down the complex fraction into two simpler fractions: one representing the real part and the other representing the imaginary part. This separation is crucial for expressing the final answer in the standard form a + bi, where a is the real part and b is the coefficient of the imaginary unit i. Moreover, this step highlights the distributive property of division over addition and subtraction, which is a key concept in algebra. By applying this property, we ensure that each term in the numerator is correctly divided by the denominator. This meticulous separation sets the stage for the final simplification step, where we will reduce each fraction to its simplest form. Thus, separating the real and imaginary parts is not just a procedural step but a strategic move that simplifies the overall expression and brings us closer to the final solution.

4. Simplify Fractions

Now, we simplify each fraction. The real part is $\frac{-4}{60}$, which simplifies to $-\frac{1}{15}$. The imaginary part is $\frac{-5\sqrt{2}i}{60}$, which simplifies to $-\frac{\sqrt{2}i}{12}$. Simplifying fractions is a crucial step in obtaining the most concise and understandable form of the expression. In this step, we focus on reducing each fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor. For the real part, $\frac{-4}{60}$, the greatest common divisor of 4 and 60 is 4. Dividing both the numerator and the denominator by 4 gives us $-\frac{1}{15}$. Similarly, for the imaginary part, $\frac{-5\sqrt{2}i}{60}$, the greatest common divisor of 5 and 60 is 5. Dividing both the numerator and the denominator by 5 gives us $-\frac{\sqrt{2}i}{12}$. This process of simplification not only makes the numbers smaller and easier to work with but also reveals the fundamental relationships between the different parts of the expression. By simplifying each fraction, we ensure that the final answer is in its most reduced form, which is a hallmark of mathematical elegance and precision. Moreover, this step reinforces the importance of basic arithmetic skills and the ability to recognize and apply the principles of fraction reduction. Thus, simplifying fractions is not just a routine task but a critical step in achieving a clear and accurate solution.

Final Answer

Combining the simplified real and imaginary parts, we get the final answer: $-\frac{1}{15} - \frac{\sqrt{2}}{12}i$. This final answer represents the simplified form of the original expression, expressed in the standard complex number form a + bi. The real part is $-\frac{1}{15}$, and the imaginary part is $-\frac{\sqrt{2}}{12}$. This result demonstrates the culmination of all the previous steps, from simplifying the square root of the negative number to separating and simplifying the real and imaginary parts. Each step has played a crucial role in transforming the initial expression into its simplest form. The final answer is not only a numerical result but also a testament to the power of algebraic manipulation and the importance of understanding complex numbers. It represents a clear and concise solution that is easy to interpret and use in further calculations. Moreover, the process of arriving at this answer has reinforced our understanding of various mathematical concepts, including the properties of square roots, the imaginary unit, fraction simplification, and the standard form of complex numbers. Thus, the final answer is not just an end point but a milestone in our mathematical journey, marking a deeper appreciation for the intricacies and elegance of complex number arithmetic.

In conclusion, by following a step-by-step approach, we successfully simplified the given expression $\frac{-4-\sqrt{-50}}{60}$ to its simplest complex number form, $-\frac{1}{15} - \frac{\sqrt{2}}{12}i$. This process underscores the importance of understanding complex numbers, simplifying square roots, and performing basic arithmetic operations with fractions. Mastering these skills is essential for more advanced mathematical studies and applications.