Solve Math Easily: 55 - 11√3 / (5 + √3)

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Guys, are you ready to dive into a math problem that might seem a bit intimidating at first glance? Don't worry, we'll break it down step-by-step to make it super easy to understand! Today, we're tackling the expression: 55 - 11√3 / 5 + √3. Our mission is to simplify this and find the solution. This problem is a perfect example of how we can use some clever techniques to make complex equations manageable. We will use a few key concepts, specifically rationalizing the denominator. This method helps us get rid of any square roots that might be hanging out in the denominator (the bottom part of the fraction). It's all about transforming the equation into a more user-friendly form. Don't worry if you're a little rusty on these concepts, we'll review them as we go along. Imagine this problem as a puzzle; each step we take is like finding a piece to make the picture clearer. We'll use tools like distribution, combining like terms, and simplifying radicals to get our answer. The goal is not just to solve the problem, but to understand the process so you can confidently tackle similar problems in the future. Mathematics can be a lot of fun if you break it down bit by bit, and that is precisely what we are going to do. Remember, the key to getting better at math is practice and persistence. So, let's put on our thinking caps and jump into the wonderful world of numbers! So, let us start, and you will realize it is very easy! Now, let us start breaking down this problem, okay?

Understanding the Problem and Strategy

Before we start crunching numbers, let's take a moment to understand what we're dealing with. The expression 55 - 11√3 / 5 + √3, at first glance, might look a bit messy, but we can clean it up. We're dealing with a combination of whole numbers, a radical (√3, which is the square root of 3), and basic arithmetic operations. The main strategy here is to simplify the expression by rationalizing the denominator and combining like terms. Rationalizing the denominator is a technique that helps us remove the radical from the denominator. This usually involves multiplying both the numerator and the denominator by a form of the denominator that eliminates the radical. We will also need to remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This will ensure we perform the calculations in the correct sequence. Our first step should involve addressing the division involving the radical. To simplify this, we need to rationalize the denominator. It's like giving a fraction a makeover so it's easier to work with. After that, we can combine any terms and, with a bit of patience, we will simplify the equation to a simple numerical value. Remember, every step should be clear and straightforward, which will ensure that we don't miss a beat along the way. Our approach will be systematic and we'll break it down into small, manageable steps. Let's get started, shall we? We will be applying a very simple step-by-step procedure to achieve this.

Rationalizing the Denominator

Alright, here's where the magic happens: rationalizing the denominator. This is a crucial step in simplifying our expression. We'll focus on the fraction part of the expression: 11√3 / (5 + √3). To rationalize the denominator (5 + √3), we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (5 + √3) is (5 - √3). The conjugate is what we use to eliminate the radical in the denominator. Why do we do this? Multiplying by the conjugate helps us use the difference of squares formula: (a + b)(a - b) = a² - b². This formula gets rid of the square root because squaring a square root gives you the original number. So, let's multiply the numerator and the denominator by (5 - √3). The numerator will become 11√3 * (5 - √3), and the denominator will be (5 + √3) * (5 - √3). This gives us 55√3 - 33 in the numerator (after distributing) and 25 - 3 in the denominator (applying the difference of squares). That simplifies to 55√3 - 33 / 22. We're making excellent progress! We've taken the messy radical from the denominator and put it into the numerator, which is much easier to deal with. Now, we can try and further simplify the expression. Simplifying can also make the equation easier to understand, which also improves your chances of solving the equation more correctly. This step is a bit like organizing your desk before you start to study; it makes the rest of the process much smoother.

Simplifying the Expression

Now that we've rationalized the denominator, let's simplify the expression further. We had 55 - (55√3 - 33) / 22. Let's focus on the fraction part again: (55√3 - 33) / 22. Notice that both terms in the numerator (55√3 and 33) are divisible by 11. We can factor out 11 from the numerator to get 11(5√3 - 3) / 22. And this can be further simplified by dividing both the numerator and the denominator by 11, which results in (5√3 - 3) / 2. Now, let's go back to our original expression: 55 - (5√3 - 3) / 2. To combine these terms, we need a common denominator. The common denominator here is 2. So, we rewrite 55 as 110/2. Now the expression becomes 110/2 - (5√3 - 3) / 2. Combining these, we have (110 - 5√3 + 3) / 2, which simplifies to (113 - 5√3) / 2. Thus, by simplifying this, it means that we have eliminated complex values, and you would be able to understand and grasp the equation. Also, note that it becomes easier to manipulate this type of equation.

Step-by-Step Solution

Here is the complete step-by-step solution for solving the math problem 55 - 11√3 / (5 + √3) and simplifying the expression:

  1. Original Expression: 55 - 11√3 / (5 + √3).
  2. Rationalize the Denominator: Multiply the numerator and denominator of 11√3 / (5 + √3) by the conjugate (5 - √3).
    • (11√3 * (5 - √3)) / ((5 + √3) * (5 - √3)) = (55√3 - 33) / (25 - 3).
  3. Simplify: This simplifies to (55√3 - 33) / 22.
  4. Factor and Simplify: Divide both numerator and denominator by 11 to get (5√3 - 3) / 2.
  5. Rewrite the original expression: 55 - (5√3 - 3) / 2
  6. Common Denominator: Rewrite 55 as 110/2 to get a common denominator.
  7. Combine Terms: (110 - (5√3 - 3)) / 2 = (110 - 5√3 + 3) / 2.
  8. Final Simplification: (113 - 5√3) / 2.

So, the solution to the expression 55 - 11√3 / (5 + √3) is (113 - 5√3) / 2. Pretty cool, right? You can see how breaking down the problem step by step makes it much less daunting.

Tips for Future Problems

Alright, you've successfully solved a math problem that might have seemed tricky at the beginning. Congratulations! To keep your math skills sharp and be ready for more challenges, here are a few tips for tackling similar problems in the future:

  • Practice Regularly: The more you practice, the more comfortable you'll become with these types of problems. Try different problems, and don't be afraid to make mistakes; they are part of the learning process.
  • Review Key Concepts: Always make sure you understand the basics, such as the order of operations, how to rationalize the denominator, and how to simplify expressions. These are the building blocks for more complex problems.
  • Break It Down: Like we did today, always break the problem down into smaller, more manageable steps. This will help you avoid making mistakes and make it easier to understand the solution.
  • Use Visual Aids: Drawing diagrams, using graphs, or writing out each step can help you see the problem more clearly. Visual aids can be especially helpful with complex math problems.
  • Check Your Work: Always double-check your work! You can do this by working backward from your answer or by using a calculator to verify your steps.
  • Seek Help When Needed: Don't hesitate to ask for help from your teacher, a friend, or an online math resource if you're struggling with a concept. Explaining it to someone else can also help you master it.
  • Stay Organized: Keep your work neat and organized. This will help you avoid making careless mistakes and make it easier to review your work.
  • Stay Positive: Believe in yourself! Math can be challenging, but with practice, you'll be able to solve more complex problems. Keep a positive attitude and celebrate your successes along the way!

Conclusion

In conclusion, we've successfully solved the mathematical problem 55 - 11√3 / (5 + √3). We did this by using the method of rationalizing the denominator, simplifying the fraction, and combining like terms. It shows that even complex problems can be solved by following a few simple steps. Remember that the key to mastering math is practice, patience, and a positive attitude. Keep practicing, and you'll find that these types of problems become easier and more manageable over time. Feel free to reach out if you have more questions; we're all in this math journey together. Keep up the great work, and happy calculating, everyone!