Rationalizing The Denominator Of 16√13 / 28√7 - A Step-by-Step Guide

Introduction to Rationalizing the Denominator

In mathematics, particularly when dealing with algebra and radical expressions, it is often necessary to rationalize the denominator. Rationalizing the denominator means eliminating any radicals (like square roots, cube roots, etc.) from the denominator of a fraction. This process simplifies the expression and makes it easier to work with in further calculations. The fundamental principle behind rationalizing the denominator is to multiply both the numerator and the denominator by a suitable factor that will eliminate the radical in the denominator. This factor is usually the radical itself or a conjugate, depending on the form of the denominator. Rationalizing denominators is not just a matter of mathematical aesthetics; it helps in standardizing mathematical expressions and facilitates comparisons and further computations. When you encounter fractions with radicals in the denominator, rationalizing them is an essential step to ensure the expression is in its simplest form. This is particularly important in more advanced mathematical contexts, such as calculus and complex analysis, where simplified expressions are crucial for problem-solving. The process is applicable to various types of radicals, including square roots, cube roots, and higher-order roots, each requiring a slightly different approach but adhering to the same core principle of eliminating the radical from the denominator. Understanding and applying this technique is a cornerstone of algebraic manipulation and is vital for success in mathematics. This process transforms the original fraction into an equivalent form that is easier to manipulate and understand, aligning with the convention of simplifying expressions for clarity and consistency. Ultimately, mastering the technique of rationalizing the denominator is an invaluable tool in the mathematician's toolkit, enhancing the ability to solve a wide range of problems with confidence and efficiency.

Problem Statement: $\frac{16 \sqrt{13}}{28 \sqrt{7}}$

We are given the expression $\frac{16 \sqrt{13}}{28 \sqrt{7}}$ and our task is to rationalize the denominator. This involves removing the square root from the denominator without changing the value of the expression. The given expression contains a fraction where both the numerator and the denominator involve radicals. Specifically, the denominator contains the term $28 \sqrt{7}$, which includes a square root. To rationalize this denominator, we need to eliminate the $\sqrt{7}$ term. We achieve this by multiplying both the numerator and the denominator by a suitable factor that will result in a rational number in the denominator. In this case, the suitable factor is $\sqrt{7}$ itself. Multiplying the denominator $28 \sqrt{7}$ by $\sqrt{7}$ will yield $28 * 7$, which is a rational number. The process of rationalizing the denominator is a standard technique in algebra, ensuring that expressions are in their simplest and most conventional form. This is particularly important when performing further operations with the expression, as it simplifies calculations and reduces the likelihood of errors. By rationalizing the denominator, we transform the expression into an equivalent form that adheres to mathematical conventions and is easier to interpret and manipulate. Furthermore, rationalizing the denominator is not just a cosmetic procedure; it is essential for various mathematical operations, such as adding or subtracting fractions with different denominators, solving equations, and simplifying complex expressions. It ensures that the expression is presented in a form that is both mathematically correct and aesthetically pleasing, making it easier to work with and understand. Mastering this technique is crucial for students and practitioners alike, forming a foundational skill for more advanced mathematical concepts.

Step-by-Step Solution

1. Simplify the coefficients:

   First, we simplify the coefficients in the fraction. The expression is $\frac{16 \sqrt{13}}{28 \sqrt{7}}$. Both 16 and 28 are divisible by 4. Dividing both by 4, we get:

   $$\frac{16 \sqrt{13}}{28 \sqrt{7}} = \frac{4 \sqrt{13}}{7 \sqrt{7}}$$

   Simplifying the coefficients is a critical first step in many algebraic problems. It reduces the complexity of the numbers involved and makes subsequent calculations easier and less prone to error. In this case, dividing both 16 and 28 by their greatest common divisor, 4, results in smaller, more manageable numbers. This step sets the stage for the rationalization process, ensuring that the numbers involved are as simple as possible. By addressing the numerical coefficients first, we streamline the overall process of simplifying the expression. It's a practice rooted in the principle of breaking down complex problems into smaller, more manageable steps, making the entire solution process more accessible and efficient. This initial simplification not only makes the numbers easier to handle but also helps in identifying potential further simplifications after the rationalization process, ensuring that the final answer is in its most reduced form.

2. Multiply by the conjugate:

   To rationalize the denominator, we need to eliminate the square root from the denominator. The denominator is $7 \sqrt{7}$. To do this, we multiply both the numerator and the denominator by $\sqrt{7}$:

   $$\frac{4 \sqrt{13}}{7 \sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$$

   Multiplying both the numerator and denominator by the same value, in this case, $\sqrt{7}$, is a key step because it doesn't change the overall value of the expression. This is because $\frac{\sqrt{7}}{\sqrt{7}}$ is equal to 1. This action is grounded in the fundamental principle of maintaining the equality of an expression while altering its form. The choice of $\sqrt{7}$ as the multiplier is strategic; it's specifically selected to cancel out the square root in the denominator when multiplied by the existing $\sqrt{7}$. This method effectively transforms the denominator into a rational number without affecting the numerical integrity of the entire fraction. The process highlights a core algebraic technique: leveraging multiplicative identities to manipulate expressions into more desirable forms. By carefully choosing the multiplier, we pave the way for simplifying the expression and presenting it in a standard format where denominators are free of radicals. This step is crucial not just for simplifying but also for facilitating further operations, such as adding or comparing fractions, where rationalized denominators are essential.

3. Multiply the numerators and denominators:

   Multiplying the numerators, we get:

   $$4 \sqrt{13} \cdot \sqrt{7} = 4 \sqrt{13 \cdot 7} = 4 \sqrt{91}$$

   Multiplying the denominators, we get:

   $$7 \sqrt{7} \cdot \sqrt{7} = 7 \cdot 7 = 49$$

   The process of multiplying the numerators and denominators is a direct application of the rules of fraction multiplication. When multiplying radicals in the numerator, we apply the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, which allows us to combine the radicals under a single square root. This step efficiently condenses the radicals into a more simplified form. In the denominator, multiplying $\sqrt{7}$ by itself results in 7, a rational number, which is then multiplied by the coefficient 7. This step is pivotal in achieving the goal of rationalizing the denominator, as it eliminates the radical from the denominator and replaces it with a rational number. The calculations are straightforward yet crucial, as they transform the original expression into a form where the denominator is rationalized, and the numerator is simplified as much as possible. By meticulously carrying out these multiplications, we move closer to the final simplified form of the expression, demonstrating the importance of careful arithmetic in algebraic manipulations. These steps exemplify how basic multiplication rules can be strategically employed to simplify complex expressions and achieve a desired mathematical form.

4. Combine the results:

   Now, we combine the results from the numerator and the denominator:

   $$\frac{4 \sqrt{91}}{49}$$

   Combining the simplified numerator and denominator is the culmination of the previous steps, presenting the expression in its rationalized form. This step clearly displays the result of the rationalization process, where the denominator is now a rational number, and the radical is confined to the numerator. This form is mathematically preferable and aligns with the standard practice of simplifying expressions. The fraction $\frac{4 \sqrt{91}}{49}$ is now in a state that is easier to interpret and use for further calculations. The radical $\sqrt{91}$ in the numerator is simplified as much as possible, as 91 does not have any perfect square factors other than 1. The combination of the simplified numerator and the rationalized denominator represents the final result of the rationalization process, achieving the objective of eliminating the radical from the denominator. This step is crucial for presenting the solution in a clear, concise, and mathematically acceptable format, underscoring the importance of meticulous execution in algebraic simplifications.

5. Final Answer:

   The final simplified expression is:

   $$\frac{4 \sqrt{91}}{49}$$

   The final answer, $\frac{4 \sqrt{91}}{49}$, represents the fully simplified form of the original expression, with the denominator rationalized and the numerator simplified. This result adheres to the mathematical convention of not leaving radicals in the denominator, providing a clear and concise solution. The process of arriving at this final answer involved several key steps: simplifying coefficients, multiplying by the conjugate to eliminate the radical in the denominator, and simplifying the resulting expression. Each step was crucial in transforming the original expression into its simplified form, demonstrating the importance of methodical and accurate algebraic manipulation. The final expression is not only mathematically correct but also aesthetically pleasing, aligning with the goal of presenting mathematical solutions in their simplest and most understandable form. This outcome underscores the effectiveness of the rationalization technique in simplifying expressions and making them easier to work with in subsequent calculations or mathematical contexts. The final answer is a testament to the power of algebraic manipulation in transforming complex expressions into simpler, more manageable forms.

Conclusion

In conclusion, we have successfully rationalized the denominator of the given expression $\frac{16 \sqrt{13}}{28 \sqrt{7}}$ to obtain the simplified form $\frac{4 \sqrt{91}}{49}$. The process involved simplifying the coefficients, multiplying both the numerator and the denominator by $\sqrt{7}$, and combining the results. This methodical approach ensures that the final expression is in its simplest form, adhering to mathematical conventions and making it easier to work with in subsequent calculations. Rationalizing the denominator is a fundamental technique in algebra, crucial for simplifying expressions and facilitating further mathematical operations. The step-by-step solution presented here provides a clear understanding of the process, highlighting the importance of careful algebraic manipulation and attention to detail. The final result not only demonstrates the practical application of this technique but also underscores the significance of simplifying expressions to their most basic form for clarity and ease of use. This skill is invaluable in various mathematical contexts, from basic algebra to more advanced topics such as calculus and complex analysis. By mastering the technique of rationalizing the denominator, students and practitioners alike can enhance their problem-solving abilities and gain a deeper appreciation for the elegance and efficiency of algebraic methods. The process not only simplifies expressions but also reinforces the core principles of mathematical manipulation, fostering a more robust understanding of mathematical concepts.