Simplifying Square Root Of -108 Expressed With I

by Admin 49 views

#include #include #include

Introduction

The realm of mathematics extends beyond the familiar territory of real numbers, venturing into the fascinating world of imaginary and complex numbers. These numbers, though seemingly abstract, play a crucial role in various fields, from electrical engineering to quantum mechanics. One of the fundamental concepts in this domain is the imaginary unit, denoted by i, which is defined as the square root of -1. This allows us to express the square roots of negative numbers in a meaningful way. In this article, we will delve into the process of expressing −108\sqrt{-108} in terms of i and simplifying it to its most basic form.

Understanding Imaginary Numbers

Before we tackle the problem at hand, it is essential to grasp the concept of imaginary numbers. Imaginary numbers are multiples of the imaginary unit i, where i is defined as −1\sqrt{-1}. This definition stems from the fact that the square of any real number is non-negative. Therefore, the square root of a negative number cannot be a real number. To address this, mathematicians introduced the concept of imaginary numbers. An imaginary number can be written in the form bi, where b is a real number and i is the imaginary unit. When we combine a real number and an imaginary number, we get a complex number, which can be expressed in the form a + bi, where a and b are real numbers. The real part of the complex number is a, and the imaginary part is b.

The significance of imaginary numbers is often underestimated, yet they form the bedrock of many mathematical and scientific theories. They are indispensable in analyzing alternating current circuits, solving differential equations, and even in the intricate equations that govern quantum mechanics. Imaginary numbers help us tackle scenarios where the traditional number line falls short. For instance, they offer solutions to equations that would otherwise be deemed unsolvable within the realm of real numbers. Grasping the essence of i as the square root of -1 allows us to manipulate and simplify expressions that involve the square roots of negative numbers, thereby expanding our mathematical toolkit.

Step-by-Step Solution for −108\sqrt{-108}

1. Factoring out -1

The first step in simplifying −108\sqrt{-108} is to recognize that we can factor out -1 from under the square root. This allows us to separate the negative sign and introduce the imaginary unit i. Mathematically, we can rewrite the expression as:

−108=−1⋅108\sqrt{-108} = \sqrt{-1 \cdot 108}

By extracting the negative one, we pave the way to express the number in terms of i, which is pivotal in dealing with square roots of negative numbers. This step is not just a mathematical manipulation; it's a conceptual bridge that connects the real number system to the complex plane, where imaginary numbers reside. Factoring out -1 is a foundational step that unveils the imaginary component hidden within the square root of a negative number.

2. Introducing the Imaginary Unit i

Now that we have factored out -1, we can use the definition of the imaginary unit i, which is −1\sqrt{-1}. Substituting this into our expression, we get:

−1⋅108=−1⋅108=i108\sqrt{-1 \cdot 108} = \sqrt{-1} \cdot \sqrt{108} = i \sqrt{108}

Here, we've replaced −1\sqrt{-1} with i, effectively bringing the imaginary unit into our simplified form. This transformation is vital because it allows us to handle the negative sign within the square root in a mathematically sound manner. By introducing i, we're acknowledging the imaginary nature of the number and setting the stage for further simplification. This step highlights the elegance of the mathematical construct of i, as it cleanly separates the imaginary component from the real component under the square root.

3. Simplifying the Square Root of 108

Next, we need to simplify 108\sqrt{108}. To do this, we look for the largest perfect square that divides 108. The prime factorization of 108 is 22â‹…332^2 \cdot 3^3. We can rewrite this as 22â‹…32â‹…32^2 \cdot 3^2 \cdot 3, which means that 3636 (626^2) is the largest perfect square factor of 108. Therefore, we can rewrite 108\sqrt{108} as:

108=36â‹…3=36â‹…3=63\sqrt{108} = \sqrt{36 \cdot 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3}

Simplifying the square root of 108 is akin to peeling away the layers to reveal its most basic form. By identifying and extracting the perfect square factor (36 in this case), we reduce the radicand (the number under the square root) to its simplest possible value. This not only makes the expression more manageable but also aligns with the mathematical principle of presenting answers in their most reduced form. The process of simplification enhances clarity and exposes the intrinsic structure of the number, making it easier to work with in subsequent calculations.

4. Combining the Terms

Now that we have simplified 108\sqrt{108} to 636\sqrt{3}, we can substitute this back into our expression:

i108=iâ‹…63=6i3i \sqrt{108} = i \cdot 6\sqrt{3} = 6i\sqrt{3}

Thus, the simplified form of −108\sqrt{-108} in terms of i is 6i36i\sqrt{3}.

By combining the terms, we arrive at the quintessential form of the expression, where the imaginary unit i and the simplified radical coexist harmoniously. This final simplification step is akin to the culmination of a mathematical journey, where each transformation brings us closer to the most elegant representation of the number. The expression 6i36i\sqrt{3} not only adheres to the conventions of mathematical notation but also provides a clear depiction of the number's imaginary nature and its relationship to the real number 3\sqrt{3}.

Alternative Methods and Insights

While the above step-by-step approach is straightforward, there are alternative methods to solve this problem. For instance, one could use a factor tree to find the prime factorization of 108 and then simplify the square root accordingly. The key is to always look for perfect square factors to simplify the radical.

Another insight is to recognize that imaginary numbers are not just abstract mathematical constructs but have practical applications in various fields. For example, in electrical engineering, imaginary numbers are used to represent alternating current (AC) circuits, where the current and voltage oscillate over time. The imaginary unit i helps engineers analyze and design these circuits by representing the phase difference between the current and voltage.

Moreover, the journey of simplifying −108\sqrt{-108} encapsulates a fundamental principle in mathematics: the quest for elegance and simplicity. Mathematical expressions are often presented in their most simplified form, not only for aesthetic reasons but also for clarity and ease of manipulation. Simplifying expressions like −108\sqrt{-108} is an exercise in mathematical problem-solving, sharpening our skills in algebraic manipulation and fostering a deeper understanding of the number system.

Conclusion

In this article, we have successfully expressed −108\sqrt{-108} in terms of i and simplified it to 6i36i\sqrt{3}. This process involved factoring out -1, introducing the imaginary unit i, simplifying the square root of 108, and combining the terms. Understanding how to work with imaginary numbers is crucial for various mathematical and scientific applications. By mastering these concepts, we expand our mathematical toolkit and gain a deeper appreciation for the richness and complexity of the number system.

Expressing −108\sqrt{-108} in terms of i and simplifying it showcases the power and elegance of complex numbers in mathematics. This exercise demonstrates how seemingly complex expressions can be broken down into manageable steps, leading to a clear and concise solution. The journey from the initial expression to the simplified form 6i36i\sqrt{3} highlights the importance of understanding the fundamental properties of imaginary numbers and the art of simplifying radicals. Mastering these techniques not only enhances our mathematical proficiency but also opens doors to more advanced concepts in algebra, calculus, and various scientific disciplines.

Thus, the exercise of simplifying −108\sqrt{-108} into 6i36i\sqrt{3} serves as a microcosm of mathematical problem-solving, emphasizing the blend of conceptual understanding and meticulous execution. The imaginary unit i is not just a symbol; it is a gateway to a realm of numbers that enriches our mathematical landscape and empowers us to tackle problems that would otherwise be insurmountable. As we continue to explore the depths of mathematics, the lessons learned from simplifying such expressions will serve as guiding principles in our quest for knowledge and understanding.