Find A, B, C, D: Expressing (√2 + √3) / (4 + √3)

by Alex Johnson 49 views

Let's dive into this fascinating mathematical problem where we aim to express the fraction (√2 + √3) / (4 + √3) in the form a√2 + b√3 + c√6 + d. This involves a bit of algebraic manipulation and a good understanding of how to rationalize denominators. We'll break it down step by step to make it clear and easy to follow.

Understanding the Problem

At its core, this problem requires us to rationalize the denominator of the given fraction. Rationalizing the denominator means getting rid of any square roots in the denominator. We achieve this by multiplying both the numerator and the denominator by the conjugate of the denominator. In our case, the denominator is (4 + √3), so its conjugate is (4 - √3). Understanding this concept is crucial, as it forms the basis for solving the problem. This process will help us transform the fraction into a form where we can easily identify the values of a, b, c, and d. Our goal is to manipulate the expression until it matches the desired format of a√2 + b√3 + c√6 + d, which will then allow us to directly read off the coefficients.

Step-by-Step Solution

  1. Rationalize the Denominator:

    To start, we multiply both the numerator and the denominator by the conjugate of the denominator (4 - √3):

    ((\√2 + √3) / (4 + √3)) * ((4 - √3) / (4 - √3))

    This step is essential because it eliminates the square root from the denominator, making the expression easier to work with. The conjugate is specifically chosen to achieve this, as multiplying (4 + √3) by (4 - √3) results in a difference of squares, which simplifies the expression.

  2. Expand the Numerator:

    Next, we expand the numerator by multiplying each term:

    (√2 + √3)(4 - √3) = 4√2 - √6 + 4√3 - 3

    This expansion involves careful multiplication of each term in the first parenthesis with each term in the second parenthesis. It's crucial to keep track of the signs and combine like terms correctly. The result is an expression that includes terms with √2, √3, √6, and a constant term.

  3. Expand the Denominator:

    Now, we expand the denominator:

    (4 + √3)(4 - √3) = 16 - 3 = 13

    This is a straightforward application of the difference of squares formula: (a + b)(a - b) = a² - b². The result is a rational number, which is exactly what we wanted when rationalizing the denominator.

  4. Combine and Simplify:

    Now, we combine the expanded numerator and denominator:

    (4√2 - √6 + 4√3 - 3) / 13

    This step involves putting the expanded numerator over the simplified denominator. We now have a single fraction that represents the original expression, but with a rational denominator.

  5. Separate the Terms:

    We separate the terms to match the form a√2 + b√3 + c√6 + d:

    (4√2 / 13) + (4√3 / 13) + (-√6 / 13) + (-3 / 13)

    This separation allows us to clearly identify the coefficients of each term. We divide each term in the numerator by the denominator, resulting in four separate fractions.

  6. Identify a, b, c, and d:

    By comparing the expression with a√2 + b√3 + c√6 + d, we find:

    • a = 4/13
    • b = 4/13
    • c = -1/13
    • d = -3/13

    This is the final step, where we directly read off the coefficients from the separated terms. Each coefficient corresponds to the value of a, b, c, or d, as defined in the original problem.

Common Mistakes to Avoid

  • Incorrectly expanding the numerator or denominator: Make sure to multiply each term properly and pay attention to signs. A small mistake in the expansion can lead to incorrect values for a, b, c, and d.
  • Forgetting to multiply both numerator and denominator by the conjugate: This is crucial for rationalizing the denominator. If you only multiply the numerator, you won't eliminate the square root in the denominator.
  • Making errors in simplification: Double-check your work when combining like terms and simplifying fractions. Simplification is a key step in arriving at the correct answer.
  • Not separating terms correctly: Ensure you separate the terms properly to match the form a√2 + b√3 + c√6 + d. This step is essential for identifying the correct coefficients.

Importance of Rationalizing the Denominator

Rationalizing the denominator is a fundamental technique in mathematics. It simplifies expressions, making them easier to work with in further calculations. It's a standard practice in algebra and calculus, and it often helps in comparing and manipulating different expressions. For example, when dealing with complex numbers or trigonometric functions, rationalizing the denominator can be a necessary step to simplify expressions and solve equations. In this specific problem, rationalizing the denominator allows us to express the fraction in the desired form, making it possible to identify the values of a, b, c, and d.

Real-World Applications

While this problem may seem purely theoretical, the techniques used here have applications in various fields. For instance, in physics, dealing with vectors and forces often involves square roots and fractions, and rationalizing the denominator can simplify calculations. In engineering, particularly in electrical engineering, similar manipulations are used when dealing with impedances and complex circuits. Moreover, in computer graphics and game development, rationalizing denominators can help in optimizing calculations involving distances and vectors, leading to more efficient code.

Conclusion

In summary, we successfully found the values of a, b, c, and d by rationalizing the denominator, expanding terms, and simplifying the expression. This problem showcases the importance of algebraic manipulation and attention to detail in mathematics. By understanding the underlying principles and practicing step-by-step solutions, you can tackle similar problems with confidence. Remember, the key is to break down the problem into manageable steps and double-check your work along the way. This methodical approach will not only help you solve problems more accurately but also deepen your understanding of the mathematical concepts involved.

For further exploration and to enhance your understanding of mathematical concepts, consider visiting resources like Khan Academy, which offers a wealth of educational materials and practice exercises. This can significantly aid in mastering various mathematical techniques and problem-solving strategies.