Rationalizing The Denominator For Sqrt(5)/sqrt(2x) A Step-by-Step Guide

Rationalizing the denominator is a fundamental technique in algebra that involves eliminating radicals from the denominator of a fraction. This process simplifies expressions and makes them easier to work with, particularly when performing further calculations or comparing different expressions. In this comprehensive guide, we will delve into the process of rationalizing the denominator for the expression $\frac{\sqrt{5}}{\sqrt{2x}}$, explore the underlying principles, and arrive at the correct answer, which is (B) $\frac{\sqrt{10x}}{2x}$. This process involves multiplying both the numerator and denominator of the fraction by a suitable expression that eliminates the radical in the denominator. The primary goal is to transform the denominator into a rational number, thereby simplifying the overall expression. This technique is particularly useful when dealing with expressions in calculus, trigonometry, and other advanced mathematical fields, as it allows for easier manipulation and simplification of complex equations and formulas. By understanding and mastering the process of rationalizing denominators, students and practitioners can significantly enhance their problem-solving skills and their ability to work with a wide range of mathematical problems involving radicals. The steps involved are straightforward but require a clear understanding of radical properties and algebraic manipulation. Let’s break down the process step by step to ensure a thorough grasp of the concept.

Understanding the Need for Rationalizing the Denominator

Before we jump into the solution, let’s understand why we rationalize denominators. The primary reason is to simplify expressions and make them easier to work with. Having a radical in the denominator can complicate further calculations, such as adding or subtracting fractions. By eliminating the radical from the denominator, we create a more standardized and manageable form. This is especially crucial in higher mathematics, where complex expressions need to be simplified for various operations and analyses. For instance, in calculus, dealing with limits and derivatives often requires simplified forms to avoid unnecessary complexity. Similarly, in physics and engineering, simplified expressions can make it easier to interpret results and make predictions. Furthermore, rationalizing the denominator allows for easier comparison of different expressions. When two expressions have denominators in different forms, comparing them directly can be challenging. By rationalizing the denominators, we can bring the expressions to a common form, making comparisons straightforward. This is particularly important in fields like statistics and data analysis, where comparing different datasets or models is a common task. In essence, rationalizing the denominator is not just a mathematical technique but a tool that enhances the clarity and usability of mathematical expressions, facilitating easier manipulation, comparison, and interpretation. The standardized form achieved through rationalization contributes significantly to the efficiency and accuracy of mathematical problem-solving across various disciplines.

Step-by-Step Solution for Rationalizing $\frac{\sqrt{5}}{\sqrt{2x}}$

To rationalize the denominator of $\frac{\sqrt{5}}{\sqrt{2x}}$, we need to eliminate the radical, which is $\sqrt{2x}$, from the denominator. Here’s a step-by-step breakdown:

Step 1: Identify the Radical in the Denominator

The first step is to clearly identify the radical term in the denominator, which in this case is $\sqrt{2x}$. This term is the one we need to eliminate to rationalize the denominator. Recognizing the specific radical term is crucial because it dictates the factor we will use to multiply both the numerator and the denominator. Misidentification at this stage can lead to an incorrect approach and an unsimplified expression. For instance, if there were multiple radical terms or a more complex expression in the denominator, accurately pinpointing the term to rationalize is paramount. This foundational step ensures that the subsequent algebraic manipulations are targeted and effective, ultimately leading to the correct simplified form of the expression. Proper identification also helps in anticipating any potential restrictions on the variable, which is particularly important in the context of square roots, where the radicand must be non-negative. Therefore, starting with a clear and accurate identification of the radical term is essential for successful rationalization.

Step 2: Determine the Rationalizing Factor

The rationalizing factor is the term we multiply both the numerator and the denominator by to eliminate the radical in the denominator. In this case, the rationalizing factor is $\sqrt{2x}$ itself. This is because multiplying $\sqrt{2x}$ by $\sqrt{2x}$ will result in $2x$, which is a rational term (assuming x is rational). Determining the correct rationalizing factor is a crucial step in the process, as it directly impacts the success of simplifying the expression. The choice of the rationalizing factor depends on the nature of the radical in the denominator. For square roots, multiplying by the same radical term is sufficient. However, for cube roots or higher-order roots, the rationalizing factor will be different. For example, to rationalize a denominator with a cube root, you would need to multiply by a factor that results in a perfect cube under the radical. Understanding how to identify and use the correct rationalizing factor is a key skill in simplifying expressions involving radicals and is essential for more advanced mathematical concepts.

Step 3: Multiply Numerator and Denominator by the Rationalizing Factor

Multiply both the numerator and the denominator by the rationalizing factor, which is $\sqrt{2x}$:

$$\frac{\sqrt{5}}{\sqrt{2x}} \times \frac{\sqrt{2x}}{\sqrt{2x}}$$

Multiplying both the numerator and the denominator by the same factor is a fundamental algebraic technique used to maintain the value of the fraction while changing its form. This is akin to multiplying the fraction by 1, which does not alter its overall value but allows us to manipulate its components. This step is crucial in the process of rationalizing the denominator because it sets up the elimination of the radical term from the denominator. By applying the rationalizing factor to both the numerator and the denominator, we ensure that the expression remains mathematically equivalent to the original while progressing towards the simplified form. This technique is not only applicable to rationalizing denominators but also to other algebraic manipulations, such as simplifying complex fractions or solving equations. Understanding and mastering this concept is essential for a strong foundation in algebra and further mathematical studies.

Step 4: Simplify the Expression

Now, simplify the expression:

Numerator: $\sqrt{5} \times \sqrt{2x} = \sqrt{10x}$

Denominator: $\sqrt{2x} \times \sqrt{2x} = 2x$

So, the expression becomes:

$$\frac{\sqrt{10x}}{2x}$$

Simplifying the expression involves performing the multiplication in both the numerator and the denominator and then reducing the fraction to its simplest form. In this case, the product of the square roots in the numerator combines to form $\sqrt{10x}$, while the product of the square roots in the denominator results in $2x$, effectively removing the radical from the denominator. The ability to simplify expressions is a fundamental skill in mathematics, as it allows for clearer understanding and easier manipulation of mathematical concepts. Simplification often involves applying the rules of arithmetic and algebra, such as combining like terms, reducing fractions, and applying the properties of exponents and radicals. It is a crucial step in problem-solving, as it reduces the complexity of the expression and makes it easier to work with in subsequent steps, whether in solving equations, evaluating functions, or performing other mathematical operations. Mastery of simplification techniques is essential for both academic success and practical applications of mathematics.

Final Answer: $\frac{\sqrt{10x}}{2x}$

Therefore, the correct answer is (B) $\frac{\sqrt{10x}}{2x}$. This process of rationalizing the denominator transforms the original expression into a form where the denominator is free of radicals, making it easier to handle in further calculations or comparisons.

Key Takeaways

• Rationalizing the denominator is a technique to eliminate radicals from the denominator of a fraction.
• The rationalizing factor is the term used to multiply both the numerator and denominator.
• Multiplying by the rationalizing factor does not change the value of the expression.
• Simplifying after multiplication is crucial to obtaining the final answer.

By following these steps, you can confidently rationalize denominators in various algebraic expressions. Remember to practice and apply this technique to different problems to solidify your understanding. This skill is not only essential for simplifying expressions but also for solving more complex mathematical problems in higher-level courses.

Practice Problems

To further enhance your understanding of rationalizing denominators, try the following practice problems:

1. Rationalize the denominator for $\frac{3}{\sqrt{7}}$
2. Rationalize the denominator for $\frac{1}{\sqrt{2} + 1}$
3. Rationalize the denominator for $\frac{\sqrt{3}}{\sqrt{5} - \sqrt{2}}$

Working through these problems will help you become more comfortable with the process and develop the skills needed to tackle more challenging expressions. Each problem presents a unique situation that requires applying the principles of rationalizing denominators in slightly different ways. For example, the second problem involves a binomial in the denominator, requiring the use of the conjugate as the rationalizing factor. The third problem combines both radical subtraction and a radical in the numerator, testing your ability to apply the steps consistently. By tackling these diverse problems, you will not only solidify your understanding of the technique but also improve your problem-solving skills and mathematical intuition. Make sure to check your answers and, if needed, review the steps outlined in this guide to identify any areas where you may need further practice. Consistent practice is key to mastering this fundamental algebraic skill.

Conclusion

In conclusion, rationalizing the denominator is a crucial algebraic technique that simplifies expressions by removing radicals from the denominator. By understanding the steps involved and practicing regularly, you can master this skill and apply it effectively in various mathematical contexts. The process not only simplifies expressions but also makes them easier to compare, manipulate, and use in further calculations. This guide has provided a comprehensive overview of rationalizing the denominator for the expression $\frac{\sqrt{5}}{\sqrt{2x}}$, demonstrating each step with clarity and providing additional context to enhance understanding. Furthermore, the inclusion of practice problems offers an opportunity to reinforce the concepts learned and develop proficiency in applying the technique. Mastery of rationalizing denominators is a valuable asset in mathematics, paving the way for success in more advanced topics and problem-solving scenarios. By consistently applying the principles and techniques discussed, you can confidently tackle expressions with radicals in the denominator and achieve simplified, more manageable forms. This skill is not just about performing a specific mathematical operation; it’s about developing a deeper understanding of algebraic manipulation and the art of simplifying complex expressions, which are essential for a strong foundation in mathematics.