Finding The Square Root Of Negative Numbers A Guide To √-98

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In the realm of mathematics, navigating complex numbers often presents intriguing challenges. Among these challenges is the task of finding the square root of negative numbers. This exploration delves into the intricacies of evaluating the expression √-98, meticulously dissecting the steps involved and illuminating the underlying principles of imaginary and complex numbers. Our journey will not only unveil the solution but also equip you with a robust understanding of how to approach similar mathematical puzzles.

At the heart of this problem lies the concept of imaginary numbers. Imaginary numbers extend the real number system by introducing the imaginary unit, denoted as 'i', which is defined as the square root of -1 (i.e., i = √-1). This seemingly simple definition unlocks a new dimension in mathematics, allowing us to grapple with the square roots of negative numbers. Before the advent of imaginary numbers, the square root of a negative number was considered undefined. However, with the introduction of 'i', we can express the square root of any negative number as a multiple of 'i'. This expansion of the number system is crucial for various fields, including electrical engineering, quantum mechanics, and signal processing, where complex phenomena often require the use of imaginary and complex numbers to be accurately modeled and understood. Grasping the fundamental nature of imaginary numbers is therefore not merely an academic exercise but a gateway to unlocking deeper mathematical and scientific insights. The ability to manipulate and interpret imaginary numbers correctly allows us to solve equations and understand relationships that would otherwise remain inaccessible within the realm of real numbers alone. This makes the understanding of 'i' and its properties indispensable for anyone delving into advanced mathematical concepts.

To find the value of √-98, we must first recognize that -98 can be expressed as the product of -1 and 98. This allows us to rewrite the expression as √(-1 * 98). By applying the property of square roots that √(a * b) = √a * √b, we can further decompose the expression into √-1 * √98. The significance of this step lies in isolating the imaginary unit, √-1, which we know is equal to 'i'. We are now left with i * √98. Our next task is to simplify √98. To do this, we need to find the prime factorization of 98. The prime factorization of 98 is 2 * 7 * 7, or 2 * 7². This means that √98 can be rewritten as √(2 * 7²). Again, applying the property of square roots, we get √2 * √7². The square root of 7² is simply 7, so we now have 7√2. Substituting this back into our original expression, i * √98, we get i * 7√2, which is commonly written as 7√2i. This step-by-step deconstruction demonstrates a systematic approach to simplifying expressions involving square roots of negative numbers, emphasizing the importance of prime factorization and the properties of square roots in navigating the realm of imaginary numbers.

Let's walk through the step-by-step solution to find the value of √-98:

  1. Rewrite √-98 as √(-1 * 98). This separates the negative sign, allowing us to deal with it using the imaginary unit.
  2. Apply the property √(a * b) = √a * √b to get √-1 * √98. This isolates the square root of -1, which is the key to introducing the imaginary unit.
  3. Recognize that √-1 = i, so the expression becomes i * √98. This is where the imaginary unit 'i' makes its appearance, transforming the problem from the realm of real numbers into the realm of complex numbers.
  4. Find the prime factorization of 98, which is 2 * 7². This step is crucial for simplifying the square root of 98.
  5. Rewrite √98 as √(2 * 7²). This step prepares us to extract the perfect square from under the radical.
  6. Apply the property √(a * b) = √a * √b again to get √2 * √7². This separates the factors under the square root, making it easier to simplify.
  7. Simplify √7² to 7, resulting in 7√2. This simplifies the radical part of the expression.
  8. Substitute 7√2 back into the expression i * √98 to get i * 7√2, which is commonly written as 7√2i. This is the final simplified form of the expression.

This meticulous step-by-step breakdown not only provides the solution but also underscores the logical progression required to solve such problems, reinforcing the understanding of imaginary numbers and the properties of square roots.

By following the steps outlined above, we've determined that the value of √-98 is 7√2i. Now, let's examine the given options to identify the correct one:

  • (A) -7√2i
  • (B) 7√2i
  • (C) 7√2i
  • (D) 7i ⋅ √2i

Upon careful inspection, we can see that option (C) matches our derived solution, 7√2i. Options (A) and (B) differ in the sign, and option (D) contains an additional imaginary unit within the square root, making them incorrect. This exercise in identifying the correct option highlights the importance of not only arriving at the solution but also verifying it against the given choices. It also underscores the necessity of paying close attention to details, such as the sign and the presence of imaginary units, to ensure the accuracy of the final answer. This step reinforces the understanding of complex numbers and the importance of precision in mathematical problem-solving.

In conclusion, the value of √-98 is indeed 7√2i. This exploration has not only unveiled the solution but has also deepened our understanding of imaginary numbers and their manipulation. By meticulously breaking down the problem into manageable steps, we've demonstrated how to approach similar mathematical challenges with confidence and precision. The journey through this problem has underscored the importance of foundational concepts, such as prime factorization and the properties of square roots, in navigating the realm of complex numbers. This knowledge is not only valuable in academic settings but also in various fields that rely on complex number theory, making the mastery of these concepts a crucial asset for any aspiring mathematician, scientist, or engineer. The ability to tackle such problems with clarity and accuracy reflects a deeper understanding of mathematical principles and an enhanced problem-solving acumen.