Rationalizing Denominator What Is The Denominator Of 7/(5√3 - 5√2)
In mathematics, rationalizing the denominator is a crucial technique used to eliminate radicals from the denominator of a fraction. This process simplifies the expression and makes it easier to work with, especially in more complex calculations. When presented with a fraction that has a radical in the denominator, such as , rationalizing the denominator involves multiplying both the numerator and the denominator by the conjugate of the denominator. This method effectively removes the square roots from the denominator, leading to a more simplified and standard form of the expression. In this article, we will delve into the step-by-step process of rationalizing the denominator of the given fraction, highlighting the underlying principles and demonstrating how to arrive at the final answer. We aim to provide a comprehensive understanding of this technique, which is fundamental in algebra and calculus.
The core concept behind rationalizing the denominator lies in the algebraic identity . This identity is particularly useful when dealing with expressions involving square roots. By multiplying the denominator by its conjugate, we transform the denominator into a difference of squares, effectively eliminating the square roots. For the given expression, the denominator is , and its conjugate is . Multiplying the original fraction by the conjugate over itself (which is equivalent to multiplying by 1) ensures that the value of the expression remains unchanged while allowing us to rationalize the denominator. This process not only simplifies the fraction but also makes it easier to compare and combine with other fractions, which is a common requirement in various mathematical problems. The ability to rationalize denominators is a fundamental skill in algebra and is essential for solving equations, simplifying expressions, and performing various mathematical operations efficiently.
The technique of rationalizing the denominator is not merely a mathematical trick; it is a systematic approach that enhances the clarity and usability of mathematical expressions. In many advanced mathematical contexts, having a rational denominator is crucial for performing operations such as adding or subtracting fractions, evaluating limits, or simplifying complex expressions. Moreover, rationalizing the denominator often reveals the underlying structure of the expression, making it easier to identify patterns and relationships. In this article, we will break down each step of the rationalization process for the fraction , providing a detailed explanation that will help you understand not only the mechanics of the process but also the rationale behind it. By mastering this technique, you will be better equipped to tackle a wide range of mathematical problems that involve radicals and fractions. The process involves algebraic manipulation, careful application of identities, and a clear understanding of how to simplify expressions, all of which are essential skills in mathematics.
Step-by-Step Rationalization
To rationalize the denominator of the fraction , we follow a systematic approach that involves identifying the conjugate of the denominator and multiplying both the numerator and the denominator by this conjugate. This method leverages the difference of squares identity, which is a fundamental concept in algebra. The process begins by recognizing the denominator as . The conjugate of this expression is obtained by changing the sign between the two terms, resulting in . Multiplying both the numerator and the denominator by this conjugate ensures that we are effectively multiplying the fraction by 1, thus preserving its value while altering its form. This is a critical step in the rationalization process, as it sets the stage for eliminating the radicals from the denominator. The subsequent steps involve performing the multiplication, simplifying the resulting expression, and arriving at the final rationalized form. Each step is crucial and must be executed carefully to avoid errors and ensure the correct outcome.
1. Identify the Conjugate
The first step in rationalizing the denominator is to identify the conjugate of the denominator. The denominator in our case is . The conjugate is formed by changing the sign between the terms, so the conjugate is . This conjugate is the key to eliminating the square roots from the denominator because when multiplied with the original denominator, it will create a difference of squares, effectively removing the radicals. Understanding how to correctly identify the conjugate is crucial, as it directly impacts the subsequent steps of the rationalization process. The conjugate always has the same terms as the original expression but with the opposite sign between them. This simple sign change is what allows us to apply the difference of squares identity, which is the foundation of this method. Without the correct conjugate, the rationalization process will not succeed in eliminating the radicals from the denominator.
2. Multiply Numerator and Denominator by the Conjugate
Next, we multiply both the numerator and the denominator of the original fraction by the conjugate. This step is crucial because it allows us to change the form of the fraction without altering its value. Multiplying by the conjugate over itself is equivalent to multiplying by 1, which preserves the fraction's original value. The expression becomes:
This multiplication sets up the application of the difference of squares identity in the denominator. The numerator will be multiplied by the conjugate, and the denominator will be multiplied by its conjugate, which will lead to the elimination of the square roots. This step is a direct application of the principle that multiplying the numerator and denominator of a fraction by the same non-zero quantity does not change the fraction's value. It is a fundamental technique in algebra and is essential for simplifying and manipulating expressions involving fractions and radicals. The careful execution of this step is vital for the success of the rationalization process.
3. Expand the Denominator
Now, we expand the denominator using the difference of squares formula, . In our case, and . Applying the formula, we get:
This step is where the magic of rationalizing the denominator happens. The difference of squares formula transforms the denominator into a form where the square roots are squared, effectively eliminating them. The expansion simplifies the denominator into a more manageable form, making the subsequent calculations easier. The difference of squares identity is a powerful tool in algebra, and its application here is a prime example of its utility. The correct application of this formula is crucial for successfully rationalizing the denominator and arriving at the simplified expression. The careful expansion of the denominator is a key step in the process.
4. Simplify the Denominator
Simplifying the expanded denominator, we have:
The denominator is now a rational number, which is the goal of rationalizing the denominator. The simplification involves squaring each term, which removes the square roots and results in whole numbers. This step demonstrates the effectiveness of using the conjugate to eliminate radicals from the denominator. The simplified denominator, 25, is a clear indication that the rationalization process is proceeding correctly. The ability to simplify expressions involving square roots and exponents is essential for mastering algebra, and this step provides a practical application of these skills. The final result of this simplification is a rational number, which is a crucial outcome in the process of rationalizing the denominator.
5. Expand the Numerator
Next, we expand the numerator:
This step involves distributing the 7 across the terms inside the parentheses. The expanded numerator is a necessary component for the final simplified fraction. The expanded form allows us to see the terms more clearly and prepare for any potential simplifications that may be possible in the final step. The distribution of the constant across the terms involving square roots is a straightforward application of the distributive property, a fundamental concept in algebra. The correct expansion of the numerator is essential for maintaining the accuracy of the expression and arriving at the correct final result.
6. Write the Rationalized Fraction
Now we write the rationalized fraction:
This step combines the simplified numerator and denominator to form the rationalized fraction. The denominator is now free of radicals, which is the primary goal of the rationalization process. The fraction is now in a more standard form, which is easier to work with in further calculations. The process of rationalizing the denominator has successfully transformed the original fraction into an equivalent form that is free of radicals in the denominator. This step is a culmination of the previous steps and represents the final form of the fraction after rationalization.
7. Simplify the Fraction (if possible)
We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
This step is an optional but often necessary part of simplifying fractions. Dividing by the greatest common divisor ensures that the fraction is in its simplest form. In this case, dividing by 5 simplifies the fraction and presents it in its most reduced form. The ability to simplify fractions is a fundamental skill in mathematics and is essential for presenting answers in the most concise and understandable manner. This final simplification step completes the rationalization process and ensures that the fraction is in its most simplified form.
Final Answer and Discussion
The denominator after rationalizing is 25. The simplified fraction is , which confirms that the denominator is indeed 25. The process of rationalizing the denominator involves multiplying the numerator and the denominator by the conjugate of the denominator, which in this case is . This technique allows us to eliminate the square roots from the denominator, making the expression easier to work with. The final answer demonstrates the effectiveness of this method and provides a clear understanding of how to rationalize denominators in similar expressions.
Conclusion
In conclusion, the denominator after rationalizing is 25. This result is achieved through the systematic process of multiplying the numerator and denominator by the conjugate of the denominator. Rationalizing the denominator is a crucial skill in algebra, allowing for the simplification of expressions and making them easier to manipulate in further calculations. The step-by-step approach outlined in this article provides a comprehensive understanding of the process, from identifying the conjugate to simplifying the final fraction. By mastering this technique, students and practitioners of mathematics can confidently tackle a wide range of problems involving radicals and fractions. The ability to rationalize denominators is not only a useful tool but also a fundamental concept in mathematical simplification.
Thus, the correct answer is (c) 25.
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