Rationalizing The Denominator Of 10 Divided By √5 + √10

Introduction

In mathematics, simplifying expressions often involves rationalizing the denominator, a process that eliminates radicals from the denominator of a fraction. This technique is crucial for performing further calculations, comparing expressions, and presenting results in a standard form. In this comprehensive guide, we will delve into the step-by-step process of rationalizing the denominator of the expression $\frac{10}{\sqrt{5} + \sqrt{10}}$. Our goal is to transform the expression into the form $a\sqrt{5} + b\sqrt{10}$, where $a$ and $b$ are rational numbers. This process not only simplifies the given expression but also provides a clear understanding of algebraic manipulation and the properties of radicals. Mastering this skill is essential for anyone studying algebra, calculus, or related fields, as it frequently appears in more complex problems. By understanding the underlying principles and techniques, you'll be better equipped to tackle a wide range of mathematical challenges. The journey through this problem will enhance your problem-solving skills and deepen your appreciation for the elegance of mathematical simplification. We'll break down each step, ensuring clarity and building a strong foundation for future algebraic endeavors. So, let's embark on this mathematical exploration together, and unlock the secrets of rationalizing denominators.

Understanding the Basics of Rationalizing the Denominator

Rationalizing the denominator is a fundamental technique in algebra used to eliminate radical expressions from the denominator of a fraction. Radicals, such as square roots, cube roots, and so on, can complicate calculations and make it difficult to compare expressions. The main objective of this process is to transform a fraction with an irrational denominator into an equivalent fraction with a rational denominator. This is typically achieved by multiplying both the numerator and the denominator by a carefully chosen expression, often the conjugate of the denominator. The conjugate of a binomial expression involving radicals (like $a + b\sqrt{c}$) is another binomial expression with the opposite sign between the terms (i.e., $a - b\sqrt{c}$). Multiplying an expression by its conjugate leverages the difference of squares formula, $(x + y)(x - y) = x^2 - y^2$, which effectively eliminates the square roots. Understanding why we rationalize denominators is crucial. It not only simplifies expressions for computational purposes but also aligns with mathematical conventions for presenting results in their simplest form. In many areas of mathematics, a fraction with a rational denominator is considered to be in its standard form, making it easier to work with and interpret. This skill is not just limited to simplifying expressions; it is a foundational concept that appears in various branches of mathematics, including calculus, trigonometry, and complex analysis. Therefore, mastering the art of rationalizing denominators is an investment in your overall mathematical proficiency. In the following sections, we will apply this understanding to the specific expression at hand, demonstrating the power and elegance of this technique.

Step-by-Step Solution: Rationalizing 10/(√5 + √10)

To rationalize the denominator of the expression $\frac{10}{\sqrt{5} + \sqrt{10}}$, we need to eliminate the radicals in the denominator. The first step involves identifying the conjugate of the denominator. In this case, the denominator is $\sqrt{5} + \sqrt{10}$, and its conjugate is $\sqrt{5} - \sqrt{10}$. The conjugate is formed by changing the sign between the terms, which is a critical step because it allows us to use the difference of squares formula. Next, we multiply both the numerator and the denominator of the fraction by this conjugate. This ensures that we are only changing the form of the expression, not its value, as we are effectively multiplying by 1. So, we have:

$$\frac{10}{\sqrt{5} + \sqrt{10}} \times \frac{\sqrt{5} - \sqrt{10}}{\sqrt{5} - \sqrt{10}}$$

Multiplying the numerators gives us $10(\sqrt{5} - \sqrt{10})$, and multiplying the denominators using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$ gives us $(\sqrt{5})^2 - (\sqrt{10})^2$. This simplifies to:

$$\frac{10(\sqrt{5} - \sqrt{10})}{5 - 10}$$

Now, we have a rational denominator. The next step is to simplify the expression further. We can simplify the denominator to -5 and distribute the 10 in the numerator:

$$\frac{10\sqrt{5} - 10\sqrt{10}}{-5}$$

Finally, we divide each term in the numerator by -5:

$$-2\sqrt{5} + 2\sqrt{10}$$

Thus, the rationalized form of the expression is $-2\sqrt{5} + 2\sqrt{10}$, which is in the desired form $a\sqrt{5} + b\sqrt{10}$, where $a = -2$ and $b = 2$. This step-by-step process clearly demonstrates how to rationalize a denominator and simplify the resulting expression.

Detailed Breakdown of the Multiplication and Simplification

In this section, we will provide a detailed breakdown of the multiplication and simplification steps involved in rationalizing the denominator of $\frac{10}{\sqrt{5} + \sqrt{10}}$. This will offer a deeper understanding of the algebraic manipulations and the properties of radicals used in the process. As established earlier, the first step is to multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{5} - \sqrt{10}$. This gives us:

$$\frac{10}{\sqrt{5} + \sqrt{10}} \times \frac{\sqrt{5} - \sqrt{10}}{\sqrt{5} - \sqrt{10}}$$

Now, let's focus on the numerator. Multiplying 10 by $(\sqrt{5} - \sqrt{10})$ involves distributing 10 across both terms:

$$10(\sqrt{5} - \sqrt{10}) = 10\sqrt{5} - 10\sqrt{10}$$

This step is a straightforward application of the distributive property. Next, we turn our attention to the denominator. Multiplying $(\sqrt{5} + \sqrt{10})$ by its conjugate $(\sqrt{5} - \sqrt{10})$ is where the difference of squares formula comes into play. Recall that $(a + b)(a - b) = a^2 - b^2$. Applying this formula, we get:

$$(\sqrt{5} + \sqrt{10})(\sqrt{5} - \sqrt{10}) = (\sqrt{5})^2 - (\sqrt{10})^2$$

Squaring the square roots simplifies the expression:

$$5 - 10 = -5$$

So, the denominator simplifies to -5. Putting the simplified numerator and denominator together, we have:

$$\frac{10\sqrt{5} - 10\sqrt{10}}{-5}$$

The final step is to divide each term in the numerator by the denominator, -5:

$$\frac{10\sqrt{5}}{-5} - \frac{10\sqrt{10}}{-5} = -2\sqrt{5} + 2\sqrt{10}$$

This completes the process of rationalizing the denominator and simplifying the expression to its final form. Each step is crucial, and understanding the underlying principles ensures accuracy and confidence in solving similar problems.

Expressing the Answer in the Form a√5 + b√10

The final step in rationalizing the denominator of $\frac{10}{\sqrt{5} + \sqrt{10}}$ is to express the answer in the specified form, $a\sqrt{5} + b\sqrt{10}$. As we've meticulously worked through the steps, we arrived at the simplified expression:

$$-2\sqrt{5} + 2\sqrt{10}$$

Now, it's straightforward to identify the values of $a$ and $b$ that fit the required form. By comparing our simplified expression with $a\sqrt{5} + b\sqrt{10}$, we can directly see that:

• $a = -2$
• $b = 2$

This means that our expression, $-2\sqrt{5} + 2\sqrt{10}$, perfectly matches the form we were asked to achieve. The coefficient of $\sqrt{5}$ is -2, and the coefficient of $\sqrt{10}$ is 2. This final confirmation ensures that we have correctly rationalized the denominator and presented the answer in the desired format. Expressing the result in this form not only satisfies the problem's requirements but also provides a clear and concise representation of the simplified expression. It highlights the importance of following instructions precisely and understanding the conventions of mathematical notation. Moreover, this process reinforces the understanding of how rational numbers interact with radicals, a fundamental concept in algebra. The ability to manipulate and simplify expressions into specific forms is a valuable skill, especially when dealing with more complex mathematical problems. Therefore, this final step is not just a formality but a crucial part of the overall problem-solving process.

Common Mistakes and How to Avoid Them

When rationalizing denominators, several common mistakes can occur, leading to incorrect answers. Understanding these pitfalls and how to avoid them is crucial for mastering this algebraic technique. One of the most frequent errors is incorrectly identifying the conjugate of the denominator. For example, if the denominator is $\sqrt{a} + \sqrt{b}$, the conjugate is $\sqrt{a} - \sqrt{b}$. A common mistake is to change the sign of only one term or to incorrectly apply the conjugate. To avoid this, always double-check that you are changing the sign between the terms, not within the terms themselves. Another common error occurs during the multiplication step. Students may forget to distribute the terms correctly when multiplying the numerator or the denominator. For instance, when multiplying $10$ by $(\sqrt{5} - \sqrt{10})$, ensure that you multiply 10 by both $\sqrt{5}$ and $\sqrt{10}$. Writing out each step explicitly can help prevent this mistake. Errors also often arise when applying the difference of squares formula, $(a + b)(a - b) = a^2 - b^2$. It's essential to remember that this formula simplifies the multiplication of conjugates, but it must be applied correctly. Misapplying this formula can lead to an incorrect denominator and subsequent errors. Simplify each term carefully and double-check your calculations. Furthermore, simplification errors can occur after the multiplication step. For example, students might incorrectly divide the terms in the numerator by the denominator or fail to simplify radicals fully. Always simplify the expression as much as possible, and remember to divide each term in the numerator by the denominator separately. Finally, a minor but important error is not expressing the answer in the required form. If the question asks for the answer in a specific format, such as $a\sqrt{5} + b\sqrt{10}$, make sure your final answer matches this form. Identifying the values of $a$ and $b$ explicitly can help avoid this mistake. By being aware of these common mistakes and practicing careful, step-by-step problem-solving, you can significantly improve your accuracy in rationalizing denominators.

Conclusion: Mastering Rationalization and Its Importance

In conclusion, mastering the technique of rationalizing denominators is a fundamental skill in algebra with far-reaching implications in mathematics and beyond. Throughout this guide, we have meticulously walked through the process of rationalizing the denominator of the expression $\frac{10}{\sqrt{5} + \sqrt{10}}$, transforming it into the simplified form $-2\sqrt{5} + 2\sqrt{10}$. This journey has highlighted the importance of understanding conjugates, the difference of squares formula, and the careful application of algebraic manipulations. Rationalizing denominators is not merely a mechanical process; it’s a crucial step in simplifying expressions, making them easier to work with and compare. It allows us to present mathematical results in a standard form, which is essential for consistency and clarity in mathematical communication. The skills acquired in this process extend beyond this specific problem. They are applicable in various areas of mathematics, including calculus, trigonometry, and complex analysis, where simplifying expressions with radicals is a common task. Furthermore, the ability to identify and correct common mistakes, such as misidentifying conjugates or incorrectly applying the difference of squares formula, is a valuable asset in any mathematical endeavor. By understanding the underlying principles and practicing diligently, you can develop confidence in your algebraic skills and enhance your problem-solving abilities. The journey of rationalizing denominators serves as a microcosm of the broader mathematical process – a blend of conceptual understanding, procedural fluency, and attention to detail. As you continue your mathematical studies, the skills and insights gained here will undoubtedly prove invaluable, empowering you to tackle more complex and challenging problems with assurance and precision. The elegance and power of mathematics lie in its ability to transform and simplify, and rationalizing denominators is a perfect illustration of this transformative power.