Express Square Root Of -147 In Terms Of I

In the realm of mathematics, complex numbers extend the familiar number system by incorporating the imaginary unit, denoted as i, which is defined as the square root of -1. This concept opens up a new dimension in mathematical operations, allowing us to work with square roots of negative numbers. In this article, we will delve into the process of expressing the square root of -147 in terms of i. This involves understanding the properties of imaginary numbers and how they interact with real numbers in complex number operations. This exploration is not just a mathematical exercise; it's a foundational step in understanding more advanced concepts in algebra, calculus, and various fields of engineering and physics. The ability to manipulate and simplify expressions involving imaginary numbers is crucial for solving equations, analyzing electrical circuits, and even understanding quantum mechanics. So, let's embark on this journey to demystify the expression of ${\sqrt{-147}}$ in terms of i, and in doing so, strengthen our grasp of complex numbers and their applications. Understanding this process will not only help in solving similar mathematical problems but also in appreciating the elegance and power of complex numbers in the broader context of mathematical and scientific disciplines. This article aims to provide a clear, step-by-step guide, ensuring that both students and enthusiasts can follow along and enhance their mathematical toolkit.

Understanding Imaginary Numbers

To express ${\sqrt{-147}}$ in terms of i, we must first grasp the fundamental concept of imaginary numbers. Imaginary numbers are multiples of the imaginary unit i, where i is defined as the square root of -1. This definition is crucial because it allows us to handle the square roots of negative numbers, which are not defined within the realm of real numbers. The introduction of i extends our number system to include complex numbers, which have both a real and an imaginary part. For example, a complex number is typically written in the form a + bi, where a represents the real part and b represents the imaginary part. The significance of i lies in its ability to bridge the gap between real and imaginary numbers, providing a cohesive framework for mathematical operations. When we encounter a square root of a negative number, such as ${\sqrt{-147}}$, we can rewrite it using i. This involves factoring out -1 from the radicand (the number under the square root) and expressing it as a product of ${\sqrt{-1}}$ and the remaining positive number. This step is essential for simplifying expressions and performing further calculations. The ability to manipulate imaginary numbers is not just an abstract mathematical skill; it has practical applications in various fields, including electrical engineering, where imaginary numbers are used to represent alternating currents and voltages. Understanding imaginary numbers is also crucial in quantum mechanics, where they play a fundamental role in describing the behavior of particles at the subatomic level. Therefore, a solid understanding of imaginary numbers is a cornerstone of mathematical and scientific literacy.

Breaking Down ${\sqrt{-147}}$

The process of expressing ${\sqrt{-147}}$ in terms of i begins with recognizing that the square root of a negative number involves imaginary components. The key is to separate the negative sign from the number under the square root. We can rewrite ${\sqrt{-147}}$ as ${\sqrt{-1 * 147}}$. This separation is a critical first step because it allows us to apply the definition of i. According to the properties of square roots, the square root of a product is the product of the square roots. Therefore, we can further rewrite the expression as ${\sqrt{-1} * \sqrt{147}}$. Now, we know that ${\sqrt{-1}}$ is defined as i, so the expression becomes i ${\sqrt{147}}$. The next step involves simplifying ${\sqrt{147}}$. To do this, we look for perfect square factors of 147. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). The prime factorization of 147 is 3 * 7 * 7, which can be written as 3 * 49. Recognizing that 49 is a perfect square (7 * 7), we can rewrite ${\sqrt{147}}$ as ${\sqrt{49 * 3}}$. Applying the property of square roots again, we get ${\sqrt{49} * \sqrt{3}}$, which simplifies to 7${\sqrt{3}}$. Now, we substitute this simplified form back into our expression with i. So, i ${\sqrt{147}}$ becomes i * 7${\sqrt{3}}$, which is typically written as 7i${\sqrt{3}}$. This final form clearly expresses the square root of -147 in terms of i, with the real and imaginary components neatly separated. This step-by-step breakdown illustrates how complex numbers can be manipulated and simplified, providing a foundation for more advanced mathematical operations.

Simplifying the Radical

The critical step in expressing ${\sqrt{-147}}$ in terms of i involves simplifying the radical, specifically ${\sqrt{147}}$. Simplification of radicals is a fundamental skill in algebra and is essential for expressing numbers in their simplest form. The goal is to identify and extract any perfect square factors from the number under the square root. As we established earlier, 147 can be factored into 3 * 49. Recognizing that 49 is a perfect square (7 * 7), we can rewrite ${\sqrt{147}}$ as ${\sqrt{49 * 3}}$. The property of square roots allows us to separate the product under the square root into the product of individual square roots: ${\sqrt{49 * 3}}$ = ${\sqrt{49}}$ * ${\sqrt{3}}$. The square root of 49 is 7, so we have 7${\sqrt{3}}$. This is the simplified form of ${\sqrt{147}}$. The process of identifying perfect square factors is crucial for simplifying radicals efficiently. It often involves looking for common factors and recognizing squares of integers. For instance, if we were simplifying ${\sqrt{72}}$, we would recognize that 72 can be factored into 36 * 2, where 36 is a perfect square (6 * 6). Thus, ${\sqrt{72}}$ would simplify to 6${\sqrt{2}}$. In the case of ${\sqrt{147}}$, the perfect square factor of 49 made the simplification straightforward. However, sometimes the perfect square factor is not immediately obvious, and it may require breaking down the number into its prime factors to identify the pairs that form perfect squares. Mastering the simplification of radicals is not only important for working with imaginary numbers but also for various other areas of mathematics, including geometry, trigonometry, and calculus. It allows for more concise and manageable expressions, making further calculations easier and more accurate. Therefore, a solid understanding of radical simplification is a valuable asset in any mathematical toolkit.

Expressing in Terms of i

Now that we have simplified ${\sqrt{147}}$ to 7${\sqrt{3}}$, the final step in expressing ${\sqrt{-147}}$ in terms of i is to combine our results. We previously established that ${\sqrt{-147}}$ can be rewritten as i${\sqrt{147}}$. We then simplified ${\sqrt{147}}$ to 7${\sqrt{3}}$. Now, we substitute the simplified radical back into the expression. So, i${\sqrt{147}}$ becomes i(7${\sqrt{3}}$). To express this in the standard form for complex numbers, we typically write the coefficient before the i. Therefore, the final expression is 7i${\sqrt{3}}$. This is the simplified form of ${\sqrt{-147}}$ in terms of i. It represents a purely imaginary number, as there is no real part (i.e., the real part is 0). Complex numbers are often written in the form a + bi, where a is the real part and bi is the imaginary part. In this case, a = 0 and b = 7${\sqrt{3}}$. The ability to express square roots of negative numbers in terms of i is fundamental to working with complex numbers. It allows us to perform algebraic operations, solve equations, and explore mathematical concepts that are not possible within the realm of real numbers alone. The expression 7i${\sqrt{3}}$ clearly demonstrates how an imaginary number can be represented and understood. It is a scalar multiple of the imaginary unit i, where the scalar is 7${\sqrt{3}}$. This representation is crucial for further mathematical manipulations, such as addition, subtraction, multiplication, and division of complex numbers. Therefore, mastering the process of expressing numbers in terms of i is a key skill in the study of complex numbers and their applications.

Conclusion

In conclusion, expressing ${\sqrt{-147}}$ in terms of i involves a series of mathematical steps that highlight the properties of imaginary and complex numbers. We began by recognizing that the square root of a negative number necessitates the use of the imaginary unit i, where i is defined as ${\sqrt{-1}}$. We then separated the negative sign from the radicand, rewriting ${\sqrt{-147}}$ as i${\sqrt{147}}$. The next crucial step was simplifying the radical ${\sqrt{147}}$. This involved identifying perfect square factors, which in this case led us to rewrite 147 as 49 * 3. Applying the properties of square roots, we simplified ${\sqrt{147}}$ to 7${\sqrt{3}}$. Finally, we combined the results to express ${\sqrt{-147}}$ as 7i${\sqrt{3}}$. This final form represents a purely imaginary number, where the real part is 0 and the imaginary part is 7${\sqrt{3}}$i. The process of expressing numbers in terms of i is not just a mathematical exercise; it is a fundamental skill in working with complex numbers. Complex numbers have wide-ranging applications in various fields, including electrical engineering, physics, and computer science. They are used to model alternating currents, analyze quantum mechanical systems, and develop algorithms for signal processing. Therefore, understanding how to manipulate and simplify expressions involving imaginary numbers is essential for students and professionals in these fields. The steps outlined in this article provide a clear and systematic approach to expressing square roots of negative numbers in terms of i. By mastering this process, one can gain a deeper appreciation for the power and elegance of complex numbers and their role in mathematics and science.