How To Rationalize The Denominator Of 2√10 / 3√11

To effectively rationalize the denominator of the expression $\frac{2 \sqrt{10}}{3 \sqrt{11}}$, we delve into the core concept of eliminating radicals from the denominator. This process is crucial in simplifying expressions and making them easier to work with, particularly in advanced mathematical calculations. The objective is to transform the denominator into a rational number without altering the overall value of the fraction. To achieve this, we must identify the appropriate fraction to multiply with the given expression. This article provides a detailed explanation of the steps involved and clarifies why multiplying by a specific fraction is the correct approach.

Understanding Rationalizing the Denominator

Rationalizing the denominator is a fundamental technique in algebra that involves removing any radical expressions, such as square roots or cube roots, from the denominator of a fraction. This process is essential for several reasons. First, it simplifies the expression, making it easier to compare with other expressions and perform further calculations. Second, it adheres to the standard mathematical convention of expressing fractions in their simplest form, where denominators are rational numbers. When we rationalize the denominator, we are essentially manipulating the fraction to achieve a more conventional and manageable form without changing its value. This involves multiplying both the numerator and the denominator by a carefully chosen expression that will eliminate the radical in the denominator. The choice of this expression depends on the form of the denominator. For instance, if the denominator contains a single square root term, we multiply by that same square root. If it contains a binomial with a square root, we multiply by its conjugate. Understanding these techniques is crucial for mastering algebraic manipulations and simplifying complex expressions.

The Importance of Rationalizing Denominators

The importance of rationalizing denominators extends beyond mere simplification. It plays a critical role in various mathematical operations and applications. Consider the scenario where you need to add or subtract fractions with irrational denominators. Rationalizing the denominators first makes it significantly easier to find a common denominator and combine the fractions. This is because working with rational denominators avoids the complexities of dealing with radicals in the denominator. Moreover, in many areas of mathematics, such as calculus and complex analysis, expressions are often required to be in their simplest form, which includes having a rational denominator. This simplifies subsequent operations, such as differentiation, integration, and evaluation of limits. In addition to these practical applications, rationalizing the denominator also serves as a good mathematical practice. It demonstrates an understanding of algebraic manipulation and the properties of radicals. It reinforces the concept of equivalent fractions and the ability to transform expressions without changing their value. By mastering this technique, students develop a deeper appreciation for the structure and elegance of mathematical expressions. Thus, rationalizing the denominator is not just a cosmetic step; it is a fundamental skill that streamlines calculations, simplifies expressions, and enhances mathematical proficiency.

Identifying the Correct Fraction for Rationalization

To identify the correct fraction to rationalize the denominator of $\frac{2 \sqrt{10}}{3 \sqrt{11}}$, we focus on the denominator, which is $3 \sqrt{11}$. Our goal is to eliminate the square root from this term. The key principle here is that multiplying a square root by itself will remove the radical, since $\sqrt{a} \cdot \sqrt{a} = a$. Therefore, to rationalize $3 \sqrt{11}$, we need to multiply it by $\sqrt{11}$. However, to maintain the value of the fraction, we must multiply both the numerator and the denominator by the same value. This is equivalent to multiplying the fraction by 1, which does not change its overall value. So, the fraction we should use is $\frac{\sqrt{11}}{\sqrt{11}}$. Multiplying the original expression by this fraction will eliminate the square root in the denominator while keeping the fraction equivalent to its original form. This approach is consistent with the fundamental rules of fraction manipulation and ensures that we are simplifying the expression in a mathematically sound way.

Why Other Options Are Incorrect

It's crucial to understand why the other options provided are not suitable for rationalizing the denominator in this specific case. Option B, $\frac{\sqrt{10}}{\sqrt{10}}$, is incorrect because it focuses on the square root in the numerator rather than the denominator. While multiplying by $\frac{\sqrt{10}}{\sqrt{10}}$ would change the numerator, it would not eliminate the radical from the denominator, which is our primary goal. Option C, $\frac{3-\sqrt{11}}{3-\sqrt{11}}$, might seem relevant because it involves the denominator's radical, but it's only applicable when the denominator is a binomial expression involving a square root. In this case, the denominator is a single term ($3\sqrt{11}$), so we don't need to use the conjugate method that this option suggests. The conjugate method is used when we have an expression like $a + \sqrt{b}$ or $a - \sqrt{b}$ in the denominator. Option D, which refers to a discussion category, is not a mathematical operation and therefore not relevant to the task of rationalizing the denominator. Understanding why these options are incorrect reinforces the importance of carefully analyzing the structure of the expression and applying the appropriate technique for rationalization. The correct choice, $\frac{\sqrt{11}}{\sqrt{11}}$, directly addresses the radical in the denominator and simplifies the expression as required.

Step-by-Step Process of Rationalizing the Denominator

The step-by-step process of rationalizing the denominator for the expression $\frac{2 \sqrt{10}}{3 \sqrt{11}}$ involves a clear and methodical approach. This ensures accuracy and a thorough understanding of the process. Here's a detailed breakdown:

1. Identify the Denominator: The first step is to clearly identify the denominator of the fraction, which in this case is $3 \sqrt{11}$. This is crucial because our focus will be on eliminating the radical from this part of the fraction.

2. Determine the Rationalizing Factor: The next step involves determining the factor that will eliminate the square root from the denominator. As discussed earlier, multiplying a square root by itself will rationalize it. Therefore, the rationalizing factor for $\sqrt{11}$ is $\sqrt{11}$ itself.

3. Construct the Rationalizing Fraction: To maintain the value of the original fraction, we need to multiply both the numerator and the denominator by the same value. This means we will multiply by the fraction $\frac{\sqrt{11}}{\sqrt{11}}$, which is equivalent to multiplying by 1.

4. Multiply the Original Fraction by the Rationalizing Fraction: Now, we multiply the original fraction by the rationalizing fraction: $ \frac{2 \sqrt{10}}{3 \sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}}

   $$$$

5. Perform the Multiplication: Multiply the numerators together and the denominators together: $ \frac{2 \sqrt{10} \times \sqrt{11}}{3 \sqrt{11} \times \sqrt{11}}

   $$$$

6. Simplify the Expression: Simplify the resulting expression. In the numerator, we have $2 \sqrt{10} \times \sqrt{11} = 2 \sqrt{10 \times 11} = 2 \sqrt{110}$. In the denominator, we have $3 \sqrt{11} \times \sqrt{11} = 3 \times 11 = 33$. So the expression becomes: $ \frac{2 \sqrt{110}}{33}

   $$$$

7. Check for Further Simplification: Finally, check if the resulting fraction can be simplified further. In this case, $\sqrt{110}$ cannot be simplified into a form that would allow us to reduce the fraction, and the fraction $\frac{2}{33}$ is already in its simplest form. Therefore, the rationalized form of the expression is $\frac{2 \sqrt{110}}{33}$.

This step-by-step approach provides a clear and concise method for rationalizing denominators, ensuring accuracy and a deeper understanding of the underlying mathematical principles.

Final Answer and Conclusion

In conclusion, to rationalize the denominator of the expression $\frac{2 \sqrt{10}}{3 \sqrt{11}}$, you should multiply the expression by the fraction $\frac{\sqrt{11}}{\sqrt{11}}$. This process eliminates the radical from the denominator, resulting in a simplified expression. The correct answer is A. Understanding and applying the technique of rationalizing denominators is a crucial skill in mathematics, particularly in algebra and calculus. It not only simplifies expressions but also facilitates further calculations and comparisons. By mastering this technique, students can enhance their mathematical proficiency and problem-solving abilities. The step-by-step process outlined in this guide provides a clear and effective method for rationalizing denominators, ensuring accuracy and a deeper understanding of the underlying principles. This skill is invaluable for simplifying complex expressions and achieving a more conventional form in mathematical notation.