Evaluating (3-√7)/(3+√7) - (3+√7)/(3-√7) Expressed As A/2 + B√7

In this detailed exploration, we embark on a mathematical journey to determine the precise value of the expression (3-√7)/(3+√7) - (3+√7)/(3-√7). This seemingly complex expression can be simplified through a series of algebraic manipulations, ultimately revealing its true form. Our goal is to express the result in the form a/2 + b√7, where a and b are constants. This article will provide a step-by-step guide, ensuring clarity and understanding for readers of all backgrounds. We will delve into the fundamental concepts of rationalizing denominators, simplifying expressions, and combining like terms. By the end of this article, you will not only grasp the solution to this particular problem but also gain a deeper appreciation for the beauty and elegance of mathematical problem-solving. Whether you are a student honing your algebra skills, a math enthusiast seeking a challenging puzzle, or simply curious about the intricacies of mathematical expressions, this guide will offer valuable insights and a clear pathway to the solution. Let's embark on this exciting mathematical adventure together and unlock the secrets hidden within this expression.

Rationalizing the Denominators: A Key Step

To effectively tackle the expression (3-√7)/(3+√7) - (3+√7)/(3-√7), the crucial first step involves rationalizing the denominators of both fractions. Rationalizing the denominator means eliminating any radical expressions from the denominator of a fraction. This is achieved by multiplying both the numerator and denominator of the fraction by the conjugate of the denominator. The conjugate of a binomial expression of the form a + b√c is a - b√c, and vice versa. This method leverages the difference of squares identity, (a + b)(a - b) = a² - b², to eliminate the square root from the denominator. Let's apply this technique to our expression. For the first fraction, (3-√7)/(3+√7), the conjugate of the denominator (3+√7) is (3-√7). We multiply both the numerator and the denominator by this conjugate: [(3-√7)/(3+√7)] * [(3-√7)/(3-√7)]. This results in [(3-√7)²]/[(3+√7)(3-√7)]. Similarly, for the second fraction, (3+√7)/(3-√7), the conjugate of the denominator (3-√7) is (3+√7). We multiply both the numerator and the denominator by this conjugate: [(3+√7)/(3-√7)] * [(3+√7)/(3+√7)]. This yields [(3+√7)²]/[(3-√7)(3+√7)]. Rationalizing the denominators in this manner is a fundamental technique in simplifying expressions involving radicals and sets the stage for further simplification and calculation. By eliminating the square roots from the denominators, we transform the expression into a more manageable form, allowing us to proceed with the next steps in finding the solution.

Expanding and Simplifying: Unveiling the Expression's Structure

Having successfully rationalized the denominators, the next crucial phase involves expanding the numerators and denominators of the resulting fractions. This step allows us to break down the squared terms and products into their individual components, revealing the underlying structure of the expression. Recall that (a - b)² = a² - 2ab + b² and (a + b)² = a² + 2ab + b². Applying these identities, we expand the numerator of the first fraction, (3-√7)², as 3² - 2(3)(√7) + (√7)² which simplifies to 9 - 6√7 + 7. Similarly, we expand the numerator of the second fraction, (3+√7)², as 3² + 2(3)(√7) + (√7)² which simplifies to 9 + 6√7 + 7. For the denominators, we use the difference of squares identity, (a + b)(a - b) = a² - b². Thus, (3+√7)(3-√7) simplifies to 3² - (√7)² = 9 - 7 = 2. Now, our expression looks significantly cleaner: (9 - 6√7 + 7)/2 - (9 + 6√7 + 7)/2. The next step is to simplify each fraction by combining like terms in the numerators. In the first fraction, 9 + 7 equals 16, giving us (16 - 6√7)/2. Similarly, in the second fraction, 9 + 7 equals 16, resulting in (16 + 6√7)/2. Expanding and simplifying the numerators and denominators is a critical step in unraveling the complexity of the expression, paving the way for the final simplification and solution. By carefully applying algebraic identities and combining like terms, we are gradually transforming the expression into a form that is easier to manipulate and understand. This methodical approach is key to navigating complex mathematical problems and arriving at the correct answer.

Combining Fractions and Simplifying Radicals: The Path to the Solution

With the fractions now having a common denominator, we proceed to the crucial step of combining the fractions and further simplifying the radicals. This involves subtracting the second fraction from the first, which is made easier by the common denominator of 2. Our expression is now in the form (16 - 6√7)/2 - (16 + 6√7)/2. To subtract the fractions, we combine the numerators while keeping the common denominator: [(16 - 6√7) - (16 + 6√7)]/2. Next, we simplify the numerator by distributing the negative sign and combining like terms. This gives us (16 - 6√7 - 16 - 6√7)/2. We observe that 16 and -16 cancel each other out, leaving us with (-6√7 - 6√7)/2. Combining the remaining terms, -6√7 - 6√7, results in -12√7. Therefore, the expression simplifies to -12√7 / 2. Finally, we divide -12√7 by 2 to obtain -6√7. This completes the simplification process, revealing the value of the original expression. Combining fractions and simplifying radicals are essential skills in algebra, allowing us to manipulate expressions and arrive at concise solutions. By carefully applying the rules of arithmetic and algebra, we have successfully navigated the complexities of the expression and found its simplified form. The result, -6√7, provides a clear and elegant answer to our initial problem. This journey through the simplification process highlights the power of algebraic techniques in unraveling mathematical puzzles and revealing the inherent beauty of mathematical expressions.

Expressing the Result in the Desired Form: a/2 + b√7

Having obtained the simplified value of the expression as -6√7, our final task is to express the result in the desired form of a/2 + b√7. This involves rewriting our simplified expression to match the given format, allowing us to identify the values of a and b. Our current result is -6√7, which can be viewed as a single term. To express this in the form a/2 + b√7, we need to identify the corresponding values for a and b. We can rewrite -6√7 as 0 + (-6)√7. Now, comparing this with a/2 + b√7, we can see that the term a/2 corresponds to 0, and the term b√7 corresponds to -6√7. To find the value of a, we set a/2 = 0, which implies that a = 0. For the value of b, we can directly see that b = -6. Therefore, the expression -6√7 can be written in the form 0/2 + (-6)√7. This fulfills the requirement of expressing the result in the form a/2 + b√7. Expressing the result in a specific form is often a crucial step in mathematical problem-solving, as it allows for clear communication of the answer and facilitates comparison with other results. By carefully matching the simplified expression with the desired format, we have successfully identified the values of a and b, completing our task. This final step underscores the importance of attention to detail and the ability to manipulate expressions to meet specific requirements. The values a = 0 and b = -6 provide a concise and complete solution to the problem, showcasing the power of algebraic simplification and expression manipulation.

Conclusion: The Final Answer and its Significance

In conclusion, after a meticulous step-by-step simplification process, we have successfully determined the value of the expression (3-√7)/(3+√7) - (3+√7)/(3-√7) and expressed it in the desired form. Starting with rationalizing the denominators, we proceeded through expanding and simplifying the numerators and denominators, combining fractions, and simplifying radicals. This journey led us to the simplified value of -6√7. Finally, we expressed this result in the form a/2 + b√7, identifying a = 0 and b = -6. Therefore, the final answer is 0/2 - 6√7, or simply -6√7. This comprehensive exploration highlights the importance of fundamental algebraic techniques in solving mathematical problems. Rationalizing denominators, simplifying expressions, and combining like terms are essential skills that enable us to navigate complex expressions and arrive at accurate solutions. The process of breaking down a complex problem into smaller, manageable steps is a valuable strategy not only in mathematics but also in various aspects of problem-solving. The solution, -6√7, is not just a numerical answer; it represents the culmination of a logical and systematic approach. It underscores the power of mathematics to reveal the underlying simplicity within seemingly complex problems. This exercise has not only provided a solution but also reinforced the importance of precision, attention to detail, and the application of fundamental mathematical principles. The journey from the initial expression to the final answer serves as a testament to the elegance and effectiveness of mathematical problem-solving. By mastering these techniques, we empower ourselves to tackle a wide range of mathematical challenges and gain a deeper appreciation for the beauty and logic of mathematics.

Therefore, the value of the expression (3-√7)/(3+√7) - (3+√7)/(3-√7) is -6√7.