Rationalizing 7/(5√3 - 5√2) What Is The Denominator?
#introduction
In the realm of mathematics, particularly in algebra and calculus, rationalizing the denominator is a fundamental technique. This process involves eliminating radicals (like square roots or cube roots) from the denominator of a fraction. By doing so, we simplify the expression, making it easier to work with and compare. This article delves into the step-by-step process of rationalizing the denominator of the expression 7/(5√3 - 5√2), providing a clear understanding of the underlying principles and techniques involved. Understanding how to rationalize denominators is crucial for various mathematical operations, including simplifying expressions, solving equations, and performing calculations in calculus and beyond. This comprehensive guide will not only show you the steps to rationalize the denominator of the given expression but also explain the reasoning behind each step, ensuring a solid grasp of the concept. By the end of this article, you will be equipped with the knowledge and skills to tackle similar problems with confidence. This technique is not just a mathematical exercise; it is a practical tool that simplifies complex expressions and makes them easier to manipulate. The ability to rationalize denominators is a cornerstone of algebraic proficiency, enabling you to solve a wide range of mathematical problems efficiently and accurately.
Understanding Rationalization
Before diving into the specific problem, let's establish a solid understanding of what rationalization entails and why it's essential. Rationalizing the denominator means transforming a fraction so that the denominator no longer contains any radicals. This is achieved by multiplying both the numerator and the denominator by a suitable expression, which we call the 'rationalizing factor.' The goal is to eliminate the radical in the denominator without changing the value of the entire fraction. This is accomplished by leveraging the property that multiplying a radical by itself (or a suitable conjugate) results in a rational number. The concept of rationalizing denominators is deeply rooted in the need for simplified and standardized mathematical expressions. When denominators contain radicals, it can complicate further calculations and comparisons. By rationalizing, we create a more manageable form that facilitates subsequent mathematical operations. For instance, when adding or subtracting fractions with radical denominators, rationalization is often a necessary first step. Moreover, in more advanced mathematical contexts, such as calculus, rationalized forms can simplify the process of differentiation and integration. The rationalizing factor is often the conjugate of the denominator. The conjugate of a binomial expression (a + b) is (a - b), and vice versa. When we multiply a binomial expression by its conjugate, we eliminate the radical terms due to the difference of squares identity: (a + b)(a - b) = a² - b². This identity is the cornerstone of many rationalization techniques, particularly when dealing with binomial denominators involving square roots. Mastering this concept not only enhances your algebraic skills but also provides a foundation for more advanced mathematical studies. The ability to rationalize denominators is a testament to mathematical elegance, transforming seemingly complex expressions into simpler, more accessible forms.
Identifying the Rationalizing Factor
In the given expression, 7/(5√3 - 5√2), the denominator is (5√3 - 5√2). To rationalize this denominator, we need to identify the appropriate rationalizing factor. As mentioned earlier, when dealing with binomial denominators involving square roots, the conjugate is the key. The conjugate of (5√3 - 5√2) is (5√3 + 5√2). Notice that we simply changed the sign between the two terms. This seemingly small change is what allows us to eliminate the radicals when we multiply. The selection of the rationalizing factor is a crucial step in the process. It's not just about changing a sign; it's about strategically choosing an expression that, when multiplied with the original denominator, will eliminate the radical terms. The conjugate serves this purpose perfectly because it exploits the difference of squares identity. By multiplying (5√3 - 5√2) by its conjugate (5√3 + 5√2), we create a scenario where the square roots are squared and thus become rational numbers. This transformation is the heart of the rationalization process. The ability to quickly and accurately identify the conjugate is a valuable skill in algebra. It demonstrates an understanding of the underlying mathematical principles and allows for efficient problem-solving. In more complex scenarios, the denominator might involve multiple terms or different types of radicals, but the fundamental principle of using a conjugate (or a similar factor) to eliminate radicals remains the same. The correct rationalizing factor ensures that the process leads to a simplified expression, devoid of radicals in the denominator. This strategic approach is what distinguishes a proficient mathematician from a novice.
Step-by-Step Rationalization Process
Now that we've identified the rationalizing factor, let's proceed with the step-by-step rationalization process. We'll multiply both the numerator and the denominator of the expression by the conjugate (5√3 + 5√2). This is a crucial step because it maintains the value of the fraction while transforming its form. Multiplying both the numerator and denominator by the same expression is equivalent to multiplying by 1, which doesn't change the fraction's value. The expression now becomes: (7 * (5√3 + 5√2)) / ((5√3 - 5√2) * (5√3 + 5√2)). The next step involves expanding both the numerator and the denominator. In the numerator, we distribute the 7 across the terms in the parentheses: 7 * 5√3 + 7 * 5√2, which simplifies to 35√3 + 35√2. In the denominator, we apply the difference of squares identity: (a - b)(a + b) = a² - b². Here, a = 5√3 and b = 5√2. So, the denominator becomes (5√3)² - (5√2)². Let's simplify the denominator further. (5√3)² equals 25 * 3, which is 75. Similarly, (5√2)² equals 25 * 2, which is 50. Therefore, the denominator simplifies to 75 - 50, which equals 25. Now, the entire expression looks like this: (35√3 + 35√2) / 25. We can simplify this fraction further by factoring out a common factor from the numerator. Both 35√3 and 35√2 have a common factor of 35. Factoring out 35, we get 35(√3 + √2) / 25. Finally, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This gives us 7(√3 + √2) / 5. The denominator after rationalization is 25. This step-by-step breakdown illustrates the methodical approach required for rationalizing denominators. Each step builds upon the previous one, leading to a simplified expression. The key is to maintain accuracy in each calculation and to understand the underlying algebraic principles.
Simplifying the Result
After performing the rationalization, we arrived at the expression 7(√3 + √2) / 5. The denominator is now 25, which is a rational number, meaning we have successfully rationalized the denominator. However, it's always a good practice to check if the resulting expression can be further simplified. In this case, the numerator is 7(√3 + √2), and the denominator is 5. There are no common factors between the numerator and the denominator that can be canceled out. The expression is in its simplest form. The final simplified expression, with a rationalized denominator, is 7(√3 + √2) / 5. The ability to simplify results is just as important as the rationalization process itself. It ensures that the expression is in its most concise and manageable form. In this instance, we examined the numerator and denominator for any common factors, but found none. This step underscores the importance of attention to detail in mathematical problem-solving. A thorough simplification process not only presents the answer in its most elegant form but also reduces the likelihood of errors in subsequent calculations. By simplifying, we make the expression easier to understand and work with, which is a fundamental goal in mathematics. The simplified result provides a clear and concise representation of the original expression with a rationalized denominator.
Conclusion
In conclusion, we have successfully rationalized the denominator of the expression 7/(5√3 - 5√2). The process involved identifying the conjugate of the denominator, multiplying both the numerator and the denominator by the conjugate, expanding the resulting expressions, simplifying the fraction, and arriving at the final simplified form. The denominator after rationalization is 25. This exercise demonstrates the importance of understanding and applying the principles of algebra, particularly the difference of squares identity and the concept of conjugates. Mastering the art of rationalizing denominators is a valuable skill in mathematics. It not only simplifies expressions but also enhances your overall problem-solving abilities. The steps involved – identifying the rationalizing factor, multiplying, expanding, simplifying – are applicable to a wide range of mathematical problems. This process is not just about getting the right answer; it's about developing a logical and methodical approach to problem-solving. The ability to rationalize denominators is a testament to mathematical fluency, allowing you to manipulate expressions with confidence and precision. By understanding the underlying principles and practicing the techniques, you can tackle similar problems with ease and achieve a deeper understanding of algebraic concepts. This comprehensive guide has provided a clear pathway to rationalizing denominators, equipping you with the knowledge and skills to excel in your mathematical endeavors. The correct answer is c) 25.
