Simplify (√(-10) * √(-5)) / √2: A Comprehensive Guide
Introduction to Complex Numbers
In the realm of mathematics, complex numbers extend the familiar number line into a two-dimensional plane, allowing us to solve equations and explore concepts that are impossible within the real number system alone. Understanding complex numbers is crucial for various fields, including engineering, physics, and computer science. At the heart of complex numbers lies the imaginary unit, denoted as i, which is defined as the square root of -1. This seemingly simple concept opens up a whole new world of mathematical possibilities, enabling us to tackle problems involving the square roots of negative numbers. The expression we aim to simplify, $\frac{\sqrt{-10} \cdot \sqrt{-5}}{\sqrt{2}}$ involves precisely this scenario, demanding a firm grasp of complex number properties and operations. Simplifying such expressions requires us to navigate the rules governing the imaginary unit and its interaction with real numbers. The beauty of complex numbers lies in their ability to provide solutions to equations that would otherwise be unsolvable, offering a more complete picture of the mathematical landscape. This exploration of complex numbers begins with understanding the basic definitions and properties, which will pave the way for simplifying more intricate expressions. This initial comprehension is not just about memorizing rules; it's about developing an intuition for how these numbers behave and interact. As we delve deeper, we will uncover the elegance and power of complex numbers in solving a variety of mathematical challenges. The journey into complex numbers is not just about finding answers; it's about expanding our mathematical horizons and gaining a deeper appreciation for the intricate relationships within the world of numbers. So, let's embark on this journey with a clear mind and a thirst for knowledge, ready to unravel the mysteries of complex numbers and their applications.
Breaking Down the Expression: $\sqrt{-10}$ and $\sqrt{-5}\
The initial step in simplifying the expression $\frac{\sqrt{-10} \cdot \sqrt{-5}}{\sqrt{2}}$ involves understanding the individual components within the numerator. Specifically, we need to address the terms $\sqrt{-10}$ and $\sqrt{-5}$, both of which involve the square roots of negative numbers. This is where the concept of imaginary numbers comes into play. Recall that the imaginary unit, denoted by i, is defined as $i = \sqrt{-1}$. This definition is the key to transforming square roots of negative numbers into expressions involving i. Focusing on $\sqrt{-10}$, we can rewrite it as $\sqrt{10 \cdot -1}$. Applying the property of square roots that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{10} \cdot \sqrt{-1}$. Since $\sqrt{-1} = i$, we can further simplify this to $\sqrt{10}i$. Now, let's turn our attention to $\sqrt{-5}$. Following a similar process, we can rewrite $\sqrt{-5}$ as $\sqrt{5 \cdot -1}$, which can be separated into $\sqrt{5} \cdot \sqrt{-1}$. Again, replacing $\sqrt{-1}$ with i, we arrive at $\sqrt{5}i$. By breaking down these initial terms and expressing them in terms of i, we've taken a significant step towards simplifying the original expression. This process highlights the importance of understanding the properties of square roots and the definition of the imaginary unit. The ability to manipulate these concepts is essential for working with complex numbers and solving related problems. As we continue, we will see how these individual simplifications combine to provide a final, concise answer. This careful, step-by-step approach is crucial for avoiding errors and ensuring a thorough understanding of the underlying principles. With these initial terms simplified, we're well-equipped to tackle the rest of the expression and uncover its true value.
Multiplying the Numerator: $\sqrt{-10} \cdot \sqrt{-5}\
Having simplified $\sqrt-10}$ to $\sqrt{10}i$ and $\sqrt{-5}$ to $\sqrt{5}i$, the next crucial step is to multiply these two expressions together. This will allow us to further consolidate the numerator of our original complex number expression. We are essentially calculating $(\sqrt{10}i) \cdot (\sqrt{5}i)$. To perform this multiplication, we can use the associative and commutative properties of multiplication. These properties allow us to rearrange and regroup the terms as followsi) \cdot (\sqrt5}i) = \sqrt{10} \cdot \sqrt{5} \cdot i \cdot i$. Now, let's focus on multiplying the square roots \cdot \sqrt5}$. Using the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, we can combine these into a single square root = \sqrt50}$. Next, we need to simplify $\sqrt{50}$. We can factor 50 as 25 * 2, and since 25 is a perfect square, we can write $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$. Now, let's turn our attention to the imaginary units$, so $i^2 = (\sqrt-1})^2 = -1$. Putting it all together, we have \cdot \sqrt{5} \cdot i \cdot i = 5\sqrt{2} \cdot (-1) = -5\sqrt{2}$. This result represents the simplified form of the numerator in our original expression. Multiplying the simplified forms of the square roots and the imaginary units has led us to a more manageable term. This careful and methodical approach ensures that we don't overlook any crucial steps and that we accurately handle the properties of both square roots and complex numbers. With the numerator now simplified, we are well-prepared to address the final step: dividing by the denominator.
Dividing by the Denominator: $\frac{-5\sqrt{2}}{\sqrt{2}}\
With the numerator simplified to $-5\sqrt2}$, we can now address the final step in simplifying the original expression$. Our expression has now been reduced to $\frac-5\sqrt{2}}{\sqrt{2}}$. This division is relatively straightforward. We have a factor of $\sqrt{2}$ in both the numerator and the denominator. This allows us to cancel out this common factor. The process is similar to simplifying fractions by dividing both the numerator and denominator by a common divisor. In this case, we can divide both the numerator and the denominator by $\sqrt{2}$}{\sqrt{2}} = -5 \cdot \frac{\sqrt{2}}{\sqrt{2}}$. Since any non-zero number divided by itself equals 1, we have $\frac{\sqrt{2}}{\sqrt{2}} = 1$. Therefore, the expression simplifies to: $-5 \cdot 1 = -5$. This final result, -5, is a real number, which might seem surprising given that we started with an expression involving square roots of negative numbers. However, this outcome highlights the way that complex numbers can interact and simplify in unexpected ways. The entire process, from breaking down the initial expression to arriving at the final answer, demonstrates the power and elegance of mathematical simplification. By carefully applying the properties of square roots, imaginary units, and fractions, we were able to transform a seemingly complex expression into a simple, real number. The key to success in these types of problems lies in a methodical approach, attention to detail, and a solid understanding of the fundamental principles of algebra and complex numbers. This example serves as a testament to the beauty of mathematics, where seemingly disparate concepts can come together to produce elegant and concise solutions. The journey of simplification has not only provided us with an answer but also reinforced our understanding of the underlying mathematical principles at play.
Final Answer: -5
After a step-by-step simplification process, we have arrived at the final answer: -5. This result succinctly captures the essence of the original expression, $\frac{\sqrt{-10} \cdot \sqrt{-5}}{\sqrt{2}}$, in its most simplified form. The journey to this answer involved several key steps, each building upon the previous one. We began by recognizing the presence of imaginary numbers, due to the square roots of negative numbers in the numerator. This led us to express $\sqrt{-10}$ and $\sqrt{-5}$ in terms of the imaginary unit, i. We then multiplied these simplified expressions together, carefully handling the $i^2$ term, which simplifies to -1. This yielded a simplified numerator. Finally, we divided the simplified numerator by the denominator, $\sqrt{2}$, and cancelled out the common factor of $\sqrt{2}$, resulting in the final answer of -5. This final answer is a real number, which showcases how operations with complex numbers can sometimes lead to real-number results. This outcome underscores the interconnectedness of different branches of mathematics and the importance of a comprehensive understanding of mathematical principles. The process of simplifying this expression has not only provided us with a numerical answer but has also reinforced our understanding of complex numbers, square roots, and algebraic manipulation. The methodical approach we employed, breaking down the problem into smaller, manageable steps, is a valuable strategy for tackling complex mathematical problems. This approach not only increases accuracy but also enhances understanding and retention. In conclusion, the simplification of $\frac{\sqrt{-10} \cdot \sqrt{-5}}{\sqrt{2}}$ to -5 serves as a clear example of the power of mathematical techniques and the elegance of mathematical solutions. The journey from a seemingly complex expression to a simple, concise answer highlights the beauty and efficiency of mathematics as a problem-solving tool.
