Solve (-1+√10)(3-2√2) Step-by-Step Solution
In the realm of mathematics, expressions involving radicals often present a unique challenge. This article delves into the detailed solution of the expression (-1+√10)(3-2√2), a fascinating problem that combines the intricacies of radical manipulation with the fundamental principles of algebra. Understanding how to simplify such expressions is not just an academic exercise; it's a crucial skill for anyone venturing into higher mathematics, physics, or engineering. This comprehensive exploration aims to break down the problem step-by-step, ensuring clarity and a deep understanding of the underlying concepts. We will cover the necessary algebraic manipulations, the properties of radicals, and the techniques required to arrive at the simplified solution. This journey will not only solve the specific problem at hand but also equip you with the tools to tackle similar mathematical challenges with confidence. The process involves applying the distributive property, simplifying radicals, and combining like terms. Each of these steps requires a careful understanding of mathematical rules and a systematic approach to avoid errors. By the end of this article, you will not only have the answer but also a solid grasp of the methodology involved.
Before we dive into the specifics of our problem, let's solidify our understanding of the core concepts at play: radicals and the distributive property. Radicals, often represented by the square root symbol (√), denote the root of a number. In our case, we have √10 and √2, both irrational numbers that, when multiplied by themselves, yield 10 and 2, respectively. The ability to manipulate radicals—simplifying them, multiplying them, and combining them—is crucial for solving expressions like (-1+√10)(3-2√2). The distributive property, a cornerstone of algebra, allows us to multiply a sum or difference by another term. It states that a(b + c) = ab + ac. In our expression, we'll use the distributive property to multiply each term in the first parenthesis by each term in the second parenthesis. This process, often referred to as the FOIL method (First, Outer, Inner, Last), ensures that every term is accounted for and multiplied correctly. Mastering these fundamentals is essential for navigating the complexities of algebraic expressions and laying a strong foundation for more advanced mathematical concepts. The interplay between radicals and the distributive property is a common theme in algebra, and this problem serves as an excellent example of how they work together.
Now, let's embark on the step-by-step solution of the expression (-1+√10)(3-2√2). Our primary tool here is the distributive property, which we'll apply meticulously to ensure accuracy. We begin by multiplying each term in the first parenthesis by each term in the second parenthesis. This can be visualized as follows:
- (-1) * 3 = -3
- (-1) * (-2√2) = 2√2
- (√10) * 3 = 3√10
- (√10) * (-2√2) = -2√(10*2) = -2√20
Combining these results, we get: -3 + 2√2 + 3√10 - 2√20. This is a crucial intermediate step where we've expanded the expression but haven't yet simplified it completely. The next step involves simplifying the radicals, particularly √20, which can be further broken down. The ability to identify and simplify radicals is key to obtaining the most concise form of the solution. This process demonstrates the power of the distributive property in expanding expressions and sets the stage for the simplification phase. By carefully applying this property, we ensure that all terms are accounted for, and we lay the groundwork for the subsequent steps in the solution.
The next crucial step in solving our expression is simplifying radicals. In our intermediate result, -3 + 2√2 + 3√10 - 2√20, we have √20 that can be further simplified. To do this, we look for perfect square factors within 20. We recognize that 20 = 4 * 5, and since 4 is a perfect square (2^2), we can rewrite √20 as √(4 * 5). Applying the property √(a * b) = √a * √b, we get √20 = √4 * √5 = 2√5. Now, substituting this back into our expression, we have: -3 + 2√2 + 3√10 - 2(2√5) = -3 + 2√2 + 3√10 - 4√5. This simplification is vital because it allows us to combine like terms, if any exist, and present the final answer in its most reduced form. Simplifying radicals not only makes the expression more manageable but also provides a clearer understanding of the numerical value it represents. This step showcases the importance of recognizing perfect square factors and applying the properties of radicals to achieve simplification. By taming the square roots, we bring the expression closer to its final, simplified form.
After simplifying the radicals, we arrive at the expression: -3 + 2√2 + 3√10 - 4√5. Now, we examine this expression to see if we can combine any like terms. Like terms are those that have the same radical component. In our case, we have √2, √10, and √5, which are all distinct radicals and cannot be combined. Therefore, the expression is already in its simplest form. The final result of (-1+√10)(3-2√2) is -3 + 2√2 + 3√10 - 4√5. This final step underscores the importance of simplifying radicals and combining like terms to arrive at the most concise solution. The journey from the initial expression to the final result demonstrates the power of algebraic manipulation and the properties of radicals. Understanding these concepts is crucial for success in mathematics and related fields. In conclusion, solving expressions involving radicals requires a systematic approach, a solid grasp of algebraic principles, and attention to detail. By breaking down the problem into manageable steps, we can navigate the complexities and arrive at the solution with confidence. This comprehensive exploration has not only provided the answer but also equipped you with the tools to tackle similar challenges in the future.
Solving (-1+√10)(3-2√2) A Step-by-Step Mathematical Solution
