Simplifying (8+√2)/(4-√7) A Step-by-Step Solution

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In the realm of mathematics, simplifying complex expressions is a fundamental skill. This article delves into the process of simplifying a specific quotient: (8+√2)/(4-√7). We will explore the steps involved in rationalizing the denominator and arriving at the correct answer. This detailed guide aims to provide a clear understanding of the techniques used and the underlying mathematical principles.

The Challenge: Simplifying the Quotient

When faced with a quotient that involves radicals in the denominator, such as (8+√2)/(4-√7), the initial instinct might be to reach for a calculator. However, to truly understand the expression and express it in its simplest form, we need to rationalize the denominator. Rationalizing the denominator is the process of eliminating the radical from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by the conjugate of the denominator.

Our primary goal is to simplify the given quotient:

(8+√2)/(4-√7)

This involves eliminating the square root in the denominator to express the quotient in a more standard and easily understandable form. The presence of the square root in the denominator makes it difficult to compare this number directly with other numbers or to use it in further calculations. Therefore, rationalizing the denominator is a crucial step in simplifying this expression.

Step-by-Step Simplification Process

To simplify the quotient (8+√2)/(4-√7), we need to follow a systematic approach. Here’s a detailed breakdown of each step:

1. Identifying the Conjugate

The first crucial step in rationalizing the denominator is identifying the conjugate of the denominator. The conjugate of a binomial expression a - b is a + b, and vice versa. In our case, the denominator is 4 - √7. Thus, the conjugate of the denominator is 4 + √7. The conjugate is essential because when multiplied by the original denominator, it eliminates the square root, which is the key to rationalization. Multiplying by the conjugate utilizes the difference of squares identity, which states that (a - b)(a + b) = a² - b². This identity is the cornerstone of this simplification process.

2. Multiplying by the Conjugate

Next, we multiply both the numerator and the denominator of the quotient by the conjugate we identified in the previous step. This ensures that we are only changing the form of the expression, not its value, as we are effectively multiplying by 1. The expression now becomes:

[(8+√2) / (4-√7)] * [(4+√7) / (4+√7)]

This step is critical because it sets up the elimination of the square root in the denominator. By multiplying both the numerator and denominator by the same value, we maintain the expression's original value while transforming its appearance.

3. Expanding the Numerator

Now, we need to expand the numerator by applying the distributive property (also known as the FOIL method). This involves multiplying each term in the first binomial by each term in the second binomial:

(8+√2) * (4+√7) = 84 + 8√7 + √24 + √2√7

This expansion yields:

32 + 8√7 + 4√2 + √14

Expanding the numerator is a straightforward application of the distributive property. Each term is carefully multiplied to ensure accuracy. This process lays the foundation for simplifying the expression further.

4. Expanding the Denominator

Similarly, we expand the denominator by multiplying (4-√7) by its conjugate (4+√7). This is where the magic of the conjugate comes into play, as it will eliminate the square root:

(4-√7) * (4+√7) = 44 + 4√7 - √74 - √7√7

This simplifies to:

16 + 4√7 - 4√7 - 7

The terms 4√7 and -4√7 cancel each other out, leaving us with:

16 - 7 = 9

The key advantage of using the conjugate is evident here. The square root term is eliminated from the denominator, resulting in a simple integer. This is a significant step toward rationalizing the denominator.

5. Simplifying the Expression

Now, we have the expanded numerator and the simplified denominator. Our expression looks like this:

(32 + 8√7 + 4√2 + √14) / 9

We examine the numerator to see if any further simplifications can be made. In this case, the terms are already in their simplest form, and there are no like terms to combine. Therefore, the simplified expression is:

(32 + 8√7 + 4√2 + √14) / 9

This step involves combining the results of the previous expansions and checking for any possible further simplifications. The expression is now in its rationalized form, with no radicals in the denominator.

Identifying the Correct Answer

Comparing our simplified expression with the given options, we find that it matches option C:

C. (32 + 8√7 + 4√2 + √14) / 9

Therefore, option C is the correct answer. The process of simplifying the quotient has led us to the correct choice, demonstrating the importance of careful and methodical steps in mathematical problem-solving.

Common Mistakes to Avoid

When simplifying expressions with radicals, it’s easy to make mistakes. Here are some common pitfalls to watch out for:

  1. Incorrectly Identifying the Conjugate: Make sure to change only the sign between the terms in the denominator. For example, the conjugate of 4 - √7 is 4 + √7, not -4 - √7.
  2. Errors in Distribution: When multiplying binomials, ensure each term is multiplied correctly. Use the FOIL method (First, Outer, Inner, Last) to avoid missing any terms.
  3. Incorrect Simplification of Radicals: Double-check that you've simplified the radicals as much as possible. For instance, √14 cannot be simplified further, but if you had √28, it should be simplified to 2√7.
  4. Forgetting to Multiply Both Numerator and Denominator: To maintain the value of the fraction, you must multiply both the numerator and the denominator by the same conjugate.
  5. Arithmetic Errors: Simple addition, subtraction, or multiplication errors can lead to an incorrect final answer. Always double-check your calculations.

Avoiding these common mistakes can significantly improve accuracy and efficiency in simplifying mathematical expressions.

Why is Rationalizing the Denominator Important?

Rationalizing the denominator is not just a mathematical exercise; it has practical implications and several advantages:

  1. Simplifying Expressions: It removes radicals from the denominator, making the expression easier to work with and understand.
  2. Facilitating Comparison: It allows for easier comparison between different expressions. When denominators are rational, it’s simpler to compare the magnitudes of different fractions.
  3. Standard Form: It presents the expression in a standard form, which is preferred in mathematical notation.
  4. Further Calculations: It makes it easier to perform additional mathematical operations, such as adding or subtracting fractions.
  5. Avoiding Ambiguity: It avoids ambiguity in certain contexts, especially when dealing with approximations or numerical computations.

The process of rationalizing the denominator ensures that expressions are in their most simplified and usable form, which is essential for both theoretical mathematics and practical applications.

Real-World Applications of Simplifying Quotients

While simplifying quotients with radicals might seem like an abstract mathematical exercise, it has real-world applications in various fields:

  1. Engineering: Engineers often encounter complex numbers and expressions in circuit analysis, signal processing, and other areas. Simplifying these expressions is crucial for accurate calculations.
  2. Physics: In physics, simplifying quotients with radicals can arise in various contexts, such as calculating energy levels in quantum mechanics or determining forces in mechanics.
  3. Computer Graphics: Computer graphics involves a lot of mathematical calculations, including those involving vectors and transformations. Simplifying quotients can help optimize these calculations.
  4. Financial Analysis: Financial analysts may encounter complex ratios and formulas that require simplification. Understanding how to rationalize denominators can aid in these calculations.
  5. Cryptography: Cryptography involves complex mathematical algorithms. Simplifying expressions is essential for efficient encryption and decryption processes.

These real-world applications highlight the importance of mastering the techniques of simplifying mathematical expressions, including rationalizing denominators.

Conclusion

Simplifying the quotient (8+√2)/(4-√7) involves rationalizing the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. This process eliminates the radical from the denominator and results in a simplified expression. By following a step-by-step approach, we correctly identified the simplified form as (32 + 8√7 + 4√2 + √14) / 9, which corresponds to option C.

This exercise underscores the significance of understanding fundamental mathematical principles and applying them methodically. Mastering these skills not only aids in solving specific problems but also builds a solid foundation for more advanced mathematical concepts and real-world applications. The ability to simplify complex expressions is a valuable asset in various fields, from engineering and physics to computer science and finance.

By avoiding common mistakes and understanding the importance of each step, you can confidently tackle similar problems and enhance your problem-solving abilities in mathematics.