Solving For A² + 1/a² When A = 3 + √8

In the realm of mathematics, particularly within algebra, we often encounter problems that require us to manipulate expressions and equations to find specific values. One such problem involves determining the value of a² + 1/a² given that a = 3 + √8. This seemingly straightforward question delves into the heart of algebraic manipulation, requiring us to understand concepts such as rationalizing denominators, squaring binomials, and simplifying expressions. This article aims to provide a comprehensive, SEO-friendly exploration of how to solve this problem, ensuring that readers not only grasp the solution but also understand the underlying principles that make it possible. Our discussion will cover the steps involved in finding the reciprocal of a, calculating a², determining 1/a², and finally, summing them up to arrive at the final answer. This journey through algebraic manipulation will highlight the elegance and precision inherent in mathematical problem-solving.

Breaking Down the Problem: Initial Steps

To effectively tackle this problem, we'll begin by dissecting it into smaller, manageable steps. Our primary goal is to find the value of a² + 1/a², but we are given a = 3 + √8. The first step involves finding 1/a, which isn't immediately obvious due to the square root in the denominator. This is where the technique of rationalizing the denominator comes into play. Rationalizing the denominator is a crucial skill in algebra, as it allows us to eliminate square roots from the denominator, making expressions easier to work with. By multiplying the numerator and denominator of 1/a by the conjugate of the denominator, we can simplify the expression and pave the way for further calculations. Once we've found 1/a, we'll move on to calculating a², which involves squaring the binomial (3 + √8). This step requires us to remember the formula for squaring a binomial: (x + y)² = x² + 2xy + y². Applying this formula correctly is essential for obtaining the correct value of a². Following the calculation of a², we'll square 1/a to find 1/a². This step will be similar to finding a², but with different values. Finally, with both a² and 1/a² in hand, we'll add them together to arrive at our final answer. Each of these steps is a building block, and mastering them will not only solve this specific problem but also enhance your overall algebraic skills.

1. Finding the Reciprocal of a: 1/a

Given that a = 3 + √8, the first crucial step in solving for a² + 1/a² is to determine the value of 1/a. This might seem straightforward, but the presence of the square root in the denominator complicates matters. To find 1/a, we essentially need to find the reciprocal of (3 + √8), which is 1/(3 + √8). However, having a square root in the denominator is not ideal in mathematical expressions. It makes it difficult to simplify further or compare with other expressions. This is where the technique of rationalizing the denominator comes into play. Rationalizing the denominator means transforming the fraction so that the denominator is a rational number, i.e., a number that can be expressed as a ratio of two integers. To rationalize the denominator of 1/(3 + √8), we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (3 + √8) is (3 - √8). This is because multiplying (3 + √8) by (3 - √8) will eliminate the square root from the denominator, as we'll see shortly. So, we multiply 1/(3 + √8) by (3 - √8)/(3 - √8). This gives us (3 - √8) in the numerator and (3 + √8)(3 - √8) in the denominator. The denominator now becomes a difference of squares: (3 + √8)(3 - √8) = 3² - (√8)² = 9 - 8 = 1. Thus, 1/a simplifies to (3 - √8)/1, which is simply 3 - √8. This step is pivotal because it transforms the expression into a more manageable form, allowing us to proceed with the subsequent calculations more easily. Understanding and applying the concept of rationalizing the denominator is a fundamental skill in algebra, and it's essential for simplifying expressions involving radicals.

2. Calculating a²

Now that we have a = 3 + √8, the next step in finding the value of a² + 1/a² is to calculate a². This involves squaring the expression (3 + √8). To do this correctly, we need to recall the formula for squaring a binomial: (x + y)² = x² + 2xy + y². This formula is a cornerstone of algebraic manipulation and is crucial for expanding expressions of this form. In our case, x = 3 and y = √8. Applying the formula, we have (3 + √8)² = 3² + 2 * 3 * √8 + (√8)². Let's break this down step by step. First, 3² is simply 9. Next, 2 * 3 * √8 equals 6√8. It's important to remember that we can't directly multiply the 6 and the 8 under the square root; we must keep the √8 as it is for now. Finally, (√8)² is simply 8, as squaring a square root cancels out the radical. So, we have (3 + √8)² = 9 + 6√8 + 8. Now, we can combine the constant terms 9 and 8, which gives us 17. Thus, a² = 17 + 6√8. This value will be crucial in the final step when we add a² and 1/a² together. The ability to correctly square binomials is a fundamental skill in algebra, and this step demonstrates its importance in simplifying expressions and solving problems. It's a skill that will be used repeatedly in more advanced mathematical topics, so mastering it now is highly beneficial.

3. Determining 1/a²

Having calculated a² and 1/a, the next logical step in our quest to find a² + 1/a² is to determine the value of 1/a². We already found that 1/a = 3 - √8. To find 1/a², we simply need to square 1/a, which means we need to calculate (3 - √8)². This again involves squaring a binomial, but this time, we have a subtraction instead of an addition. The formula for squaring a binomial still applies, but we need to be careful with the signs. The formula is (x - y)² = x² - 2xy + y². In our case, x = 3 and y = √8. Applying the formula, we have (3 - √8)² = 3² - 2 * 3 * √8 + (√8)². Let's break this down as we did before. First, 3² is 9. Next, -2 * 3 * √8 equals -6√8. Notice the negative sign here, which is crucial. Finally, (√8)² is 8. So, we have (3 - √8)² = 9 - 6√8 + 8. Now, we combine the constant terms 9 and 8, which gives us 17. Thus, 1/a² = 17 - 6√8. This value is the counterpart to the a² we calculated earlier, and together, they will allow us to find the final answer. The process of squaring binomials, whether with addition or subtraction, is a fundamental skill in algebra, and this step reinforces its importance. Understanding how signs affect the outcome is particularly crucial, and this example highlights that aspect.

The Final Calculation: a² + 1/a²

With the values of a² and 1/a² now determined, we are finally ready to calculate a² + 1/a². We found that a² = 17 + 6√8 and 1/a² = 17 - 6√8. To find their sum, we simply add the two expressions together: (17 + 6√8) + (17 - 6√8). This step is where we see the elegance of the previous manipulations. Notice that the terms involving the square root, +6√8 and -6√8, are additive inverses. This means they will cancel each other out when added together. This cancellation is not a coincidence; it's a direct result of rationalizing the denominator and squaring the conjugate expressions. The only terms that remain are the constants: 17 + 17. Adding these together, we get 34. Therefore, a² + 1/a² = 34. This final answer is a clean, rational number, which is a satisfying result given the presence of square roots in the initial problem. This problem demonstrates how seemingly complex expressions can be simplified through careful algebraic manipulation. The key steps of rationalizing the denominator, squaring binomials, and recognizing additive inverses are all crucial techniques in algebra, and this problem provides a good example of how they work together to solve a problem. The ability to break down a problem into smaller, manageable steps and apply the appropriate algebraic techniques is a hallmark of mathematical proficiency.

Summarizing the Solution Process

To recap, we started with a = 3 + √8 and were asked to find the value of a² + 1/a². The solution involved several key steps, each building upon the previous one. First, we found 1/a by rationalizing the denominator of 1/(3 + √8). This involved multiplying the numerator and denominator by the conjugate of the denominator, (3 - √8), which gave us 1/a = 3 - √8. Next, we calculated a² by squaring (3 + √8) using the formula for squaring a binomial: (x + y)² = x² + 2xy + y². This gave us a² = 17 + 6√8. Then, we found 1/a² by squaring 1/a, which is (3 - √8)². Again using the binomial squaring formula, this time for (x - y)², we found 1/a² = 17 - 6√8. Finally, we added a² and 1/a² together: (17 + 6√8) + (17 - 6√8). The terms involving the square root canceled each other out, leaving us with 17 + 17 = 34. Therefore, a² + 1/a² = 34. This problem highlights the importance of several key algebraic skills, including rationalizing denominators, squaring binomials, and simplifying expressions. It also demonstrates how breaking down a complex problem into smaller, more manageable steps can make it easier to solve. The ability to recognize patterns and apply appropriate techniques is crucial in mathematics, and this problem provides a good example of how these skills can be used to arrive at a solution. Understanding these concepts and practicing these techniques will undoubtedly improve your problem-solving abilities in algebra and beyond.

Conclusion: The Beauty of Algebraic Manipulation

In conclusion, the problem of finding the value of a² + 1/a² given a = 3 + √8 is a testament to the power and elegance of algebraic manipulation. By systematically applying techniques such as rationalizing the denominator, squaring binomials, and simplifying expressions, we were able to transform a seemingly complex problem into a straightforward calculation. The key takeaway from this problem is not just the final answer, but the process itself. Each step, from finding the reciprocal of a to calculating a² and 1/a², showcases the beauty of mathematical reasoning and the precision required to solve such problems. Moreover, this problem underscores the importance of mastering fundamental algebraic skills. The ability to rationalize denominators, expand binomials, and simplify expressions are not just isolated techniques; they are building blocks that form the foundation of more advanced mathematical concepts. By practicing these skills and understanding their applications, students can develop a deeper appreciation for the beauty and power of mathematics. The solution to this problem also highlights the interconnectedness of different mathematical concepts. The conjugate of a binomial, the formula for squaring a binomial, and the properties of square roots all come together to provide a clear and concise solution. This interconnectedness is a hallmark of mathematics, and recognizing these connections can lead to a more intuitive understanding of the subject. Ultimately, the problem of finding a² + 1/a² given a = 3 + √8 is more than just a mathematical exercise; it's an opportunity to appreciate the elegance, precision, and interconnectedness of mathematics. By mastering the techniques involved and understanding the underlying principles, students can unlock the beauty and power of algebra and apply these skills to solve a wide range of mathematical problems.