Rationalizing Denominators How To Simplify (2√(10))/(3√(11))

When dealing with fractions in mathematics, especially those involving radicals (square roots, cube roots, etc.), it's often necessary to rationalize the denominator. This process eliminates radicals from the denominator, resulting in a simplified and more standard form of the expression. This article will delve into the concept of rationalizing denominators, specifically addressing the question of which fraction to multiply $\frac{2 \sqrt{10}}{3 \sqrt{11}}$ by to achieve this. We'll explore the underlying principles, the steps involved, and the rationale behind this technique.

Understanding Rationalizing the Denominator

The core idea behind rationalizing the denominator is to manipulate the fraction so that no radical terms appear in the denominator. This is primarily a matter of convention, as it makes it easier to compare and perform operations with such fractions. Having a rational denominator also simplifies further calculations and analysis. Rationalizing the denominator is essential for simplifying expressions and making them easier to work with. Consider the fraction $\frac{1}{\sqrt{2}}$. While mathematically valid, it's common practice to eliminate the square root from the denominator. We achieve this by multiplying both the numerator and denominator by $\sqrt{2}$, resulting in $\frac{\sqrt{2}}{2}$. This transformed fraction is equivalent to the original but has a rational denominator. This process not only adheres to mathematical conventions but also streamlines calculations and facilitates comparisons between different expressions. In essence, rationalizing the denominator makes mathematical expressions cleaner and more manageable. The primary goal is to eliminate any radical expressions, such as square roots or cube roots, from the denominator of a fraction. This simplification technique is not about changing the value of the fraction; rather, it's about rewriting the fraction in an equivalent form that adheres to standard mathematical practice. This practice is particularly crucial in algebra and calculus, where simplified expressions lead to easier manipulation and problem-solving. The process often involves multiplying the numerator and the denominator by a specific term that will eliminate the radical in the denominator. This term is carefully chosen based on the form of the denominator, whether it's a simple square root, a binomial expression involving square roots, or a higher-order radical.

The Case of $\frac{2 \sqrt{10}}{3 \sqrt{11}}$

Now, let's focus on the specific expression $\frac{2 \sqrt{10}}{3 \sqrt{11}}$. Our objective is to rationalize the denominator, which currently contains the term $3 \sqrt{11}$. To eliminate the square root, we need to multiply both the numerator and the denominator by a factor that will result in a perfect square under the radical in the denominator. The key lies in identifying the appropriate fraction to multiply with. We need to consider the options provided and determine which one will effectively remove the square root from the denominator without altering the value of the entire expression. Understanding the properties of square roots is crucial here. Recall that $\sqrt{a} \cdot \sqrt{a} = a$. This principle is the foundation of rationalizing denominators when dealing with simple square root terms. In this case, we want to multiply $\sqrt{11}$ by something that will give us 11 under the radical. This points us towards the correct approach to solve the problem. By carefully selecting the fraction, we can transform the original expression into an equivalent form with a rational denominator, making it easier to work with in subsequent mathematical operations or comparisons. The process highlights the importance of recognizing and applying fundamental algebraic principles to simplify complex expressions effectively.

Analyzing the Options

We are presented with four options, each representing a fraction that could potentially be used to rationalize the denominator of $\frac{2 \sqrt{10}}{3 \sqrt{11}}$. Let's analyze each option:

• A. $\frac{\sqrt{10}}{\sqrt{10}}$: Multiplying by this fraction would affect the numerator by introducing $\sqrt{10} \cdot \sqrt{10} = 10$, but it would not directly address the $\sqrt{11}$ in the denominator. While multiplying by $\frac{\sqrt{10}}{\sqrt{10}}$ doesn't change the value of the expression (since it's equivalent to multiplying by 1), it doesn't serve the purpose of rationalizing the denominator in this specific case. It would introduce a rational term in the numerator, but the core issue of having a square root in the denominator would remain unresolved. Therefore, this option is not the correct approach for this particular problem. The goal is to eliminate the square root in the denominator, and this fraction doesn't achieve that objective.

• B. $\frac{\sqrt{11}}{\sqrt{11}}$: This option appears promising. Multiplying the denominator $3 \sqrt{11}$ by $\sqrt{11}$ would result in $3 \cdot 11 = 33$, effectively eliminating the square root. This is the key insight in rationalizing denominators – multiplying a square root by itself yields a rational number. Let's examine why this works: When we multiply $\sqrt{11}$ by $\sqrt{11}$, we are essentially calculating $\sqrt{11} * \sqrt{11}$, which equals $\sqrt{11 * 11}$, or $\sqrt{121}$. The square root of 121 is 11, a rational number. This transformation successfully removes the square root from the denominator, which is the primary goal of rationalizing. Therefore, option B seems like the correct choice as it directly addresses the issue of the square root in the denominator.

• C. $\frac{2-\sqrt{10}}{2-\sqrt{10}}$: This option involves a more complex expression. Multiplying by this fraction would not directly eliminate the $\sqrt{11}$ in the denominator. This type of fraction is typically used when the denominator contains a binomial expression involving a square root, such as $2 + \sqrt{10}$, where we would multiply by its conjugate. The conjugate is formed by changing the sign between the terms (in this case, $2 - \sqrt{10}$). However, in our original problem, the denominator is simply $3\sqrt{11}$, a monomial with a single square root term. Multiplying by the conjugate technique is not necessary or appropriate here. This option would unnecessarily complicate the expression without addressing the fundamental goal of rationalizing the denominator by removing the square root from the denominator. Therefore, option C is not the correct approach for this problem.

The Correct Answer and Justification

Based on our analysis, the correct answer is B. $\frac{\sqrt{11}}{\sqrt{11}}$. Multiplying both the numerator and the denominator of $\frac{2 \sqrt{10}}{3 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ will rationalize the denominator. Let's perform the multiplication to illustrate this:

$\frac{2 \sqrt{10}}{3 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} = \frac{2 \sqrt{10} \cdot \sqrt{11}}{3 \sqrt{11} \cdot \sqrt{11}}$

Now, we simplify:

Numerator: $2 \sqrt{10} \cdot \sqrt{11} = 2 \sqrt{10 \cdot 11} = 2 \sqrt{110}$

Denominator: $3 \sqrt{11} \cdot \sqrt{11} = 3 \cdot 11 = 33$

Therefore, the result is:

$\frac{2 \sqrt{110}}{33}$

As we can see, the denominator is now a rational number (33), and we have successfully rationalized the denominator. This demonstrates the effectiveness of multiplying by $\frac{\sqrt{11}}{\sqrt{11}}$ in this specific case. This process not only simplifies the expression but also adheres to the standard mathematical convention of avoiding radicals in the denominator.

Conclusion

In conclusion, to rationalize the denominator of $\frac{2 \sqrt{10}}{3 \sqrt{11}}$, you should multiply the expression by the fraction $\frac{\sqrt{11}}{\sqrt{11}}$. This approach effectively eliminates the square root from the denominator, resulting in a simplified and more conventional form of the expression. Understanding the principles behind rationalizing denominators is crucial for simplifying expressions and performing further mathematical operations. This technique is a fundamental skill in algebra and is widely used in various mathematical contexts. By mastering this concept, students can confidently manipulate expressions involving radicals and achieve accurate and simplified results. This process enhances mathematical fluency and problem-solving capabilities. Remember, the key is to identify the appropriate factor that will eliminate the radical in the denominator without altering the value of the original expression. In this case, multiplying by the square root present in the denominator does the trick, showcasing a practical application of square root properties in simplifying algebraic expressions.