Solving A Math Problem: Finding A + B Value

Hey math enthusiasts! Let's dive into a cool algebra problem. The question is: If 1/(1/2 - 1/√5) = a + b√5, what's the value of a + b? This type of problem is pretty common in mathematics, testing your skills in simplifying expressions and working with surds (square roots). Don't worry, it's not as scary as it looks! We'll break it down step-by-step, making sure you understand every little detail. This is the kind of question that shows up in math competitions or in your regular school tests, so it's super useful to know how to tackle it. The key here is to rationalize the denominator and isolate the values of 'a' and 'b'. Once we've done that, finding a + b is a piece of cake. So, grab your pencils, and let's get started! We'll go through the entire process clearly and concisely. Remember, the more you practice, the better you get, so consider this a fun challenge to sharpen your math skills. This journey will involve some basic algebraic manipulations, but the core idea is all about simplifying the given expression to match the form a + b√5. We'll start with the initial expression and work our way towards the answer. The problem gives us a good opportunity to practice our skills with fractions and radicals, ensuring we can apply these techniques correctly. Getting the right answer is satisfying, but understanding the process is even more rewarding, so let's make sure we understand it fully.

Rationalizing the Denominator and Simplifying the Expression

The first crucial step in solving this problem is to rationalize the denominator. What does that mean? Essentially, we want to get rid of the square root in the denominator. Here's how we do it: start with the given expression, 1/(1/2 - 1/√5). To rationalize, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate is found by changing the sign between the terms in the denominator. In our case, the denominator is (1/2 - 1/√5), so its conjugate is (1/2 + 1/√5). Multiplying both the numerator and denominator by this conjugate gives us: (1 * (1/2 + 1/√5)) / ((1/2 - 1/√5) * (1/2 + 1/√5)). Now, let's simplify. The numerator becomes 1/2 + 1/√5. For the denominator, remember the difference of squares formula: (x - y)(x + y) = x² - y². Applying this, we get (1/2)² - (1/√5)² = 1/4 - 1/5. Now let's simplify further. The numerator remains 1/2 + 1/√5. Calculate the denominator: 1/4 - 1/5 = 5/20 - 4/20 = 1/20. So our expression now looks like (1/2 + 1/√5) / (1/20). To get rid of the fraction in the denominator, multiply the entire expression by 20: 20 * (1/2 + 1/√5) = 10 + 20/√5. The next step is to simplify the term involving the square root. To do this, multiply the numerator and denominator of 20/√5 by √5. This gives us (20 * √5) / (√5 * √5) = (20√5) / 5 = 4√5. Thus, our simplified expression is 10 + 4√5. When we get to the final step, we must not make any mistakes. Also, in this step, we must follow the rules of mathematics.

Determining the Values of 'a' and 'b'

At this stage of our solution, we have simplified the expression to 10 + 4√5. Now, recall that the original equation stated that 1/(1/2 - 1/√5) = a + b√5. We've just shown that 1/(1/2 - 1/√5) simplifies to 10 + 4√5. Therefore, we can say that a + b√5 = 10 + 4√5. To find the values of 'a' and 'b', we just need to compare the coefficients of the terms in both expressions. In the expression a + b√5, 'a' is the constant term, and 'b' is the coefficient of √5. Similarly, in the simplified expression 10 + 4√5, 10 is the constant term, and 4 is the coefficient of √5. Comparing these, we can see that a = 10 and b = 4. We have successfully determined the values of a and b. Now that we have our values, we can use these values to solve the final question. Remember that the key to this step is to properly compare the coefficients of the terms. Understanding this part will make solving similar problems much easier in the future. The ability to quickly and accurately identify these values is crucial. The correct determination of 'a' and 'b' is the stepping stone to finding the final answer.

Calculating the Value of a + b

Alright, we're in the home stretch! We've determined that a = 10 and b = 4. The original question asked us to find the value of a + b. This is super simple! We just add the values of 'a' and 'b' together: a + b = 10 + 4 = 14. And there you have it! The value of a + b is 14. Congratulations, guys, we solved the problem! This is one of the best examples of where to apply the knowledge we have learned. Make sure you review the steps to ensure you understand all the details of the question. Also, this exercise can be applied to any level of question if the problem requires similar steps. Keep in mind that math problems often build on basic principles. The more you understand these principles, the easier complex problems become. This solution provides a clear path to understand and solve this type of question. Always double-check your calculations, especially during exams, to minimize any mistakes. Understanding the overall process is more important than just getting the right answer. This process helps to prepare you for more complex calculations and problem-solving. Feel confident with this problem, and try other questions! You can do it.

Conclusion

So, to sum it up, we started with a complex fraction, rationalized the denominator, simplified it, compared the coefficients, and finally found that a + b = 14. This problem beautifully illustrates the power of mathematical manipulation. It shows how we can transform a seemingly complicated expression into a simple solution using fundamental principles. By practicing these steps, you'll become more confident and skilled in solving similar problems. Math is a journey, and each problem you solve is a step forward. Don't be discouraged by challenging problems; embrace them as opportunities to learn and grow. Keep exploring, keep practicing, and you'll be amazed at what you can achieve. Remember, consistency is key. Make it a habit to solve math problems regularly to keep your skills sharp. That's all for today, folks! Hope you enjoyed this math journey. Keep practicing, and you'll become math masters in no time. Now go out there and solve some more equations! You got this!