Rationalizing The Denominator Of 15/(√7+√2) A Detailed Guide
In the realm of mathematics, rationalizing the denominator is a crucial technique used to simplify expressions involving radicals in the denominator. This process eliminates the radical from the denominator, making the expression easier to work with and understand. In this comprehensive guide, we will delve into the intricacies of rationalizing the denominator for the expression 15/(√7+√2) and simplifying it to its simplest form. This exploration will not only enhance your understanding of this fundamental mathematical concept but also equip you with the skills to tackle similar problems with confidence.
Understanding the Concept of Rationalizing the Denominator
Before we embark on the step-by-step solution, it's essential to grasp the underlying principle of rationalizing the denominator. The goal is to eliminate any radicals, such as square roots or cube roots, from the denominator of a fraction. This is achieved by multiplying both the numerator and denominator by a carefully chosen expression that will eliminate the radical in the denominator. For expressions involving the sum or difference of square roots, we employ the concept of conjugates. The conjugate of an expression of the form a + b is a - b, and vice versa. Multiplying an expression by its conjugate eliminates the radical due to the difference of squares identity: (a + b)(a - b) = a² - b². By understanding this fundamental principle, we set the stage for effectively rationalizing the denominator in our given expression.
Identifying the Conjugate and Multiplying
To rationalize the denominator of 15/(√7+√2), our first step involves identifying the conjugate of the denominator. The denominator is √7 + √2, so its conjugate is √7 - √2. We then multiply both the numerator and denominator of the fraction by this conjugate. This step is crucial because it sets the stage for eliminating the radicals in the denominator. Multiplying by the conjugate ensures that when we apply the difference of squares identity, the square roots will be eliminated, resulting in a rational denominator. This process maintains the value of the expression while transforming its form, making it easier to work with and simplify further.
Applying the Distributive Property and Simplifying
Now that we've multiplied both the numerator and denominator by the conjugate, we proceed to apply the distributive property in both the numerator and the denominator. In the numerator, we multiply 15 by (√7 - √2), resulting in 15√7 - 15√2. In the denominator, we multiply (√7 + √2) by (√7 - √2). This is where the magic of conjugates comes into play. Applying the difference of squares identity, (a + b)(a - b) = a² - b², we get (√7)² - (√2)². This simplifies to 7 - 2, which equals 5. By applying the distributive property and the difference of squares identity, we've successfully eliminated the radicals from the denominator, paving the way for further simplification.
Step-by-Step Solution: Rationalizing the Denominator
Let's walk through the process step-by-step to ensure clarity and understanding. This methodical approach will solidify your grasp of the technique and enable you to apply it confidently in various mathematical scenarios. Each step is carefully explained to provide a comprehensive understanding of the underlying logic and the mathematical principles at play. By following this detailed solution, you will gain a deeper appreciation for the elegance and efficiency of rationalizing the denominator.
Step 1: Identify the Conjugate
The denominator of the given expression is √7 + √2. The conjugate of this expression is √7 - √2. Remember, the conjugate is formed by simply changing the sign between the two terms. Identifying the conjugate correctly is the cornerstone of the rationalization process, as it sets the stage for eliminating the radicals in the denominator. This seemingly simple step is crucial for the subsequent steps and the overall success of the simplification.
Step 2: Multiply by the Conjugate
Multiply both the numerator and the denominator of the fraction by the conjugate: (√7 - √2). This step is the heart of the rationalization process. By multiplying both the numerator and the denominator by the same expression, we are essentially multiplying the fraction by 1, which doesn't change its value. However, this strategic multiplication transforms the expression into a form where the denominator can be rationalized. This technique is a powerful tool in simplifying expressions and making them easier to work with.
15/(√7 + √2) * (√7 - √2)/(√7 - √2)
Step 3: Apply the Distributive Property
Apply the distributive property in the numerator: 15 * (√7 - √2) = 15√7 - 15√2. This step expands the numerator, preparing it for further simplification. The distributive property is a fundamental algebraic principle that allows us to multiply a single term by a group of terms within parentheses. Applying it correctly ensures that each term in the parentheses is multiplied by the term outside, maintaining the integrity of the expression.
In the denominator, multiply (√7 + √2) by (√7 - √2). This is where the difference of squares identity comes into play. This identity is a cornerstone of algebra and is particularly useful in rationalizing denominators. By recognizing and applying this identity, we can efficiently eliminate the radicals in the denominator, simplifying the expression significantly.
Step 4: Simplify the Denominator
Using the difference of squares identity, (a + b)(a - b) = a² - b², we get: (√7)² - (√2)² = 7 - 2 = 5. This step demonstrates the power of using conjugates. By multiplying the denominator by its conjugate, we have successfully eliminated the square roots, resulting in a rational number. This simplification is a key objective of rationalizing the denominator and makes the expression much easier to handle.
Step 5: Simplify the Entire Expression
Now we have the expression (15√7 - 15√2) / 5. Notice that 15 is a common factor in the numerator. Factoring out 15, we get: 15(√7 - √2) / 5. This step highlights the importance of recognizing common factors in simplifying expressions. Factoring out common factors allows us to reduce the expression to its simplest form, making it easier to understand and work with.
Divide both the numerator and the denominator by 5: (15/5)(√7 - √2) = 3(√7 - √2). This final simplification step completes the process of rationalizing the denominator. By dividing both the numerator and the denominator by their greatest common factor, we arrive at the simplest form of the expression. This final form is not only more concise but also easier to interpret and use in further calculations.
The Simplified Result
Therefore, the simplified form of 15/(√7+√2) after rationalizing the denominator is 3(√7 - √2). This result is not only mathematically accurate but also aesthetically pleasing. By rationalizing the denominator, we have transformed the expression into a form that is free of radicals in the denominator, making it easier to work with and understand. This technique is a valuable tool in various mathematical contexts and is essential for simplifying expressions and solving equations.
Importance of Rationalizing the Denominator
Rationalizing the denominator is not merely a mathematical exercise; it holds significant practical importance. It simplifies expressions, making them easier to compare, combine, and evaluate. In various mathematical contexts, such as calculus and algebra, expressions with rationalized denominators are often preferred for further calculations. Moreover, rationalizing the denominator is crucial in standardizing mathematical notation, ensuring clarity and consistency in mathematical communication. By understanding and applying this technique, we enhance our ability to work with mathematical expressions effectively and efficiently.
Additional Examples and Practice Problems
To further solidify your understanding, let's explore additional examples and practice problems. These examples will showcase the versatility of the rationalization technique and help you develop your problem-solving skills. By working through a variety of problems, you will gain confidence in your ability to tackle any expression that requires rationalizing the denominator.
Example 1: Rationalize the denominator for 4/(√3 - 1)
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Identify the conjugate of the denominator: The conjugate of √3 - 1 is √3 + 1.
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Multiply both the numerator and denominator by the conjugate:
4/(√3 - 1) * (√3 + 1)/(√3 + 1) -
Apply the distributive property:
Numerator: 4 * (√3 + 1) = 4√3 + 4 Denominator: (√3 - 1) * (√3 + 1) = (√3)² - (1)² = 3 - 1 = 2 -
Simplify the expression:
(4√3 + 4) / 2 = 2√3 + 2
Practice Problem 1: Rationalize the denominator for 10/(√5 + √3)
Example 2: Rationalize the denominator for 1/(√2 + √3)
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Identify the conjugate of the denominator: The conjugate of √2 + √3 is √2 - √3.
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Multiply both the numerator and denominator by the conjugate:
1/(√2 + √3) * (√2 - √3)/(√2 - √3) -
Apply the distributive property:
Numerator: 1 * (√2 - √3) = √2 - √3 Denominator: (√2 + √3) * (√2 - √3) = (√2)² - (√3)² = 2 - 3 = -1 -
Simplify the expression:
(√2 - √3) / -1 = √3 - √2
Practice Problem 2: Rationalize the denominator for 6/(2 - √3)
By working through these examples and practice problems, you will strengthen your understanding of rationalizing the denominator and develop the skills to apply this technique effectively in various mathematical situations. Remember, practice is key to mastering any mathematical concept. The more you practice, the more confident and proficient you will become.
Conclusion
In conclusion, rationalizing the denominator is a fundamental technique in mathematics that simplifies expressions and makes them easier to work with. By understanding the concept of conjugates and applying the difference of squares identity, we can effectively eliminate radicals from the denominator of a fraction. The simplified form of 15/(√7+√2) is 3(√7 - √2). This comprehensive guide has provided a step-by-step solution, additional examples, and practice problems to help you master this essential skill. Remember, practice makes perfect, so continue to explore and apply this technique to various mathematical problems. By doing so, you will not only enhance your mathematical proficiency but also gain a deeper appreciation for the beauty and elegance of mathematics.
