Finding The Value Of A² + B² Given A = 5 + 2/6 And B = 1/a
In the realm of mathematics, we often encounter problems that require us to manipulate equations and expressions to arrive at a solution. One such problem involves finding the value of a² + b² given specific values for a and b. This article delves into the step-by-step process of solving this problem, providing a clear and comprehensive explanation for readers of all backgrounds.
Understanding the Problem
At the heart of this mathematical challenge lies the need to determine the value of a² + b². We are given that a equals 5 plus 2/6, and b is defined as the reciprocal of a (1/a). Before we can calculate a² + b², we must first simplify the expression for a and then find the value of b. This foundational understanding sets the stage for a smooth and logical progression towards the final answer.
Simplifying a
The initial step involves simplifying the expression for a, which is given as 5 + 2/6. To accomplish this, we need to combine the whole number (5) with the fraction (2/6). This process requires finding a common denominator. We can express 5 as a fraction with a denominator of 6 by multiplying both the numerator and denominator by 6, resulting in 30/6. Now, we can add the two fractions: 30/6 + 2/6. Adding fractions with the same denominator is straightforward: we simply add the numerators and keep the denominator. Therefore, 30/6 + 2/6 equals 32/6. This fraction can be further simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2. Simplifying 32/6 gives us 16/3. Thus, we have determined that a is equal to 16/3. This simplification is a crucial step, as it makes subsequent calculations much easier to handle.
Determining the Value of b
Now that we have simplified the value of a, we can move on to finding the value of b. Remember that b is defined as the reciprocal of a, which means b = 1/a. Since a = 16/3, the reciprocal of a is simply the inverse of this fraction. To find the reciprocal of a fraction, we swap the numerator and the denominator. Therefore, the reciprocal of 16/3 is 3/16. This means that b is equal to 3/16. Having both a and b expressed as simple fractions sets us up perfectly for the final calculation.
Calculating a² + b²
With the values of a and b determined, we can now calculate a² + b². Recall that a = 16/3 and b = 3/16. To find a², we need to square the fraction 16/3. Squaring a fraction means multiplying it by itself: (16/3) * (16/3). When multiplying fractions, we multiply the numerators together and the denominators together. So, (16/3) * (16/3) = (16 * 16) / (3 * 3) = 256/9. Therefore, a² = 256/9. Next, we need to find b², which means squaring the fraction 3/16. Similarly, we multiply the fraction by itself: (3/16) * (3/16). Multiplying the numerators gives us 3 * 3 = 9, and multiplying the denominators gives us 16 * 16 = 256. So, (3/16) * (3/16) = 9/256. Therefore, b² = 9/256.
Adding a² and b²
Now that we have the values of a² and b², we can add them together to find the final answer. We have a² = 256/9 and b² = 9/256. To add these fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of 9 and 256. Since 9 and 256 have no common factors other than 1, their least common multiple is simply their product: 9 * 256 = 2304. Now we need to express both fractions with the common denominator of 2304. To convert 256/9 to a fraction with a denominator of 2304, we multiply both the numerator and the denominator by 256: (256/9) * (256/256) = (256 * 256) / 2304 = 65536/2304. To convert 9/256 to a fraction with a denominator of 2304, we multiply both the numerator and the denominator by 9: (9/256) * (9/9) = (9 * 9) / 2304 = 81/2304. Now we can add the fractions: 65536/2304 + 81/2304. Adding the numerators, we get 65536 + 81 = 65617. The denominator remains the same. Therefore, a² + b² = 65617/2304. This fraction is already in its simplest form, as 65617 and 2304 have no common factors other than 1.
Conclusion
In summary, we have successfully found the value of a² + b² given that a = 5 + 2/6 and b = 1/a. The process involved simplifying the expression for a, finding the value of b, calculating a² and b², and finally adding them together. Through careful step-by-step calculations, we arrived at the final answer: a² + b² = 65617/2304. This problem exemplifies how a methodical approach and a solid understanding of basic mathematical principles can lead to the solution of seemingly complex problems. This exploration reinforces the importance of mastering fundamental concepts in mathematics, which serve as building blocks for tackling more advanced challenges. The ability to simplify expressions, find reciprocals, square fractions, and add fractions with different denominators are all crucial skills that are honed through practice and application. This example serves as a valuable exercise for anyone looking to strengthen their mathematical abilities and problem-solving skills.
This detailed walkthrough not only provides the solution but also aims to enhance understanding of the underlying mathematical concepts, making it a valuable learning resource for students and anyone interested in mathematics. By breaking down the problem into manageable steps and explaining the reasoning behind each step, this article makes the solution accessible and understandable to a wide audience.
