Rationalize Denominator And Simplify Expression -7/(6-2√3)

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Introduction

In mathematics, particularly when dealing with expressions involving radicals, rationalizing the denominator is a crucial technique. This process eliminates radicals from the denominator of a fraction, making it easier to work with and often leading to a more simplified form. In this comprehensive article, we will delve into the process of rationalizing the denominator of the expression $\frac{-7}{6-2 \sqrt{3}}$. We will break down each step, providing a clear and detailed explanation to ensure a thorough understanding. This skill is fundamental in algebra and calculus, and mastering it will significantly enhance your ability to manipulate and simplify mathematical expressions. The given expression, $\frac{-7}{6-2 \sqrt{3}}$, presents a classic scenario where rationalization is necessary. The presence of the square root in the denominator complicates further operations and comparisons. By rationalizing the denominator, we transform the expression into an equivalent form that is often more manageable and aesthetically pleasing. This article aims to not only provide a step-by-step solution but also to elucidate the underlying principles and reasons behind each step, thereby fostering a deeper comprehension of the mathematical concepts involved. Understanding the rationale behind rationalizing denominators is as important as knowing the method itself. It allows you to apply this technique confidently in various mathematical contexts and problems. So, let's embark on this journey of simplifying expressions and mastering the art of rationalizing denominators.

Understanding Rationalizing the Denominator

Before we tackle the specific expression, let's first understand what rationalizing the denominator actually means and why it's so important. At its core, rationalizing the denominator is the process of eliminating any radical expressions (like square roots, cube roots, etc.) from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a carefully chosen expression, which we often call the conjugate. But why do we bother with this process? The primary reason is to simplify the expression and make it easier to work with. Expressions with rational denominators are generally considered to be in a more standard form. This makes it simpler to compare fractions, perform arithmetic operations, and further manipulate the expression in algebraic or calculus problems. Furthermore, rationalizing the denominator often leads to a more concise and elegant form of the expression. In many mathematical contexts, having a simplified expression is crucial for clear communication and accurate calculations. For instance, when dealing with limits in calculus or simplifying complex numbers, a rationalized denominator can significantly ease the process. The concept of a conjugate is central to rationalizing denominators, especially when the denominator is a binomial involving a square root. The conjugate of an expression like a + b√c is a - b√c, and vice versa. The key property of conjugates is that when they are multiplied, the result is a difference of squares, which eliminates the square root. This is the fundamental principle we exploit when rationalizing denominators. Understanding these underlying concepts and the reasons behind them is essential for mastering the technique of rationalizing denominators. It not only allows you to solve problems efficiently but also deepens your understanding of mathematical principles.

Identifying the Conjugate

In order to rationalize the denominator of the given expression, $\frac{-7}{6-2 \sqrt{3}}$, the first crucial step is to identify the conjugate of the denominator. As we discussed, the conjugate plays a pivotal role in eliminating the radical from the denominator. In our case, the denominator is 6 - 2√3. To find its conjugate, we simply change the sign between the two terms. Therefore, the conjugate of 6 - 2√3 is 6 + 2√3. This seemingly simple step is the foundation of the entire process. The conjugate is the key that unlocks the simplification of the expression. By multiplying the denominator by its conjugate, we will be able to eliminate the square root, as the product will result in a difference of squares. It's important to understand why this works. When we multiply a binomial expression by its conjugate, we are essentially applying the algebraic identity (a - b)(a + b) = a² - b². This identity is particularly useful when dealing with expressions involving square roots because squaring a square root eliminates the radical. The correct identification of the conjugate is paramount. A mistake at this stage will lead to an incorrect rationalization and a failure to simplify the expression. Therefore, it's always a good practice to double-check the conjugate before proceeding with the multiplication. The ability to quickly and accurately identify conjugates is a valuable skill in algebra and is frequently used in various mathematical problems, not just in rationalizing denominators. So, with the conjugate of our denominator identified as 6 + 2√3, we are now ready to move on to the next step: multiplying both the numerator and the denominator by this conjugate.

Multiplying by the Conjugate

Now that we have identified the conjugate of the denominator as 6 + 2√3, the next step in rationalizing the denominator is to multiply both the numerator and the denominator of the original expression, $\frac{-7}{6-2 \sqrt{3}}$, by this conjugate. This process is crucial because multiplying by the conjugate is the mechanism by which we eliminate the radical from the denominator. Mathematically, this can be represented as follows:

frac−76−2sqrt3timesfrac6+2sqrt36+2sqrt3\\frac{-7}{6-2 \\sqrt{3}} \\times \\frac{6+2 \\sqrt{3}}{6+2 \\sqrt{3}}

It's essential to multiply both the numerator and the denominator by the same expression. This is equivalent to multiplying the entire fraction by 1, which does not change its value, only its form. This principle is fundamental in many algebraic manipulations. Let's break down the multiplication process. In the numerator, we have -7 multiplied by (6 + 2√3). This involves distributing the -7 across both terms:

-7 * 6 = -42

-7 * 2√3 = -14√3

So, the numerator becomes -42 - 14√3. Now, let's focus on the denominator. We are multiplying (6 - 2√3) by its conjugate (6 + 2√3). As discussed earlier, this is where the magic happens. We apply the difference of squares identity, (a - b)(a + b) = a² - b²:

(6 - 2√3)(6 + 2√3) = 6² - (2√3)²

6² is simply 36.

(2√3)² is 2² * (√3)² = 4 * 3 = 12.

Therefore, the denominator becomes 36 - 12 = 24. The multiplication by the conjugate has successfully eliminated the square root from the denominator. This step demonstrates the power and elegance of using conjugates in simplifying expressions. It's a technique that is widely applicable in various mathematical contexts. With the multiplication complete, our expression now looks like this:

frac−42−14sqrt324\\frac{-42 - 14\\sqrt{3}}{24}

We are one step closer to the final simplified form. The next step involves simplifying this resulting expression, which we will explore in detail in the following section.

Simplifying the Expression

After multiplying the numerator and the denominator by the conjugate, we arrived at the expression $\frac{-42 - 14\sqrt{3}}{24}$. The next critical step in rationalizing the denominator is to simplify this expression. Simplification is a fundamental aspect of mathematical problem-solving, as it presents the expression in its most concise and manageable form. In this case, simplification involves identifying common factors in the numerator and the denominator and then dividing both by these factors. Looking at the numerator, -42 - 14√3, we can see that both terms have a common factor of -14. Similarly, the denominator, 24, also shares factors with -14. The greatest common factor (GCF) of -42, -14, and 24 is 2. However, we can actually factor out -14 from the numerator, which will further simplify the expression. Factoring out -14 from the numerator, we get:

-14(3 + √3)

Now, our expression looks like this:

frac−14(3+sqrt3)24\\frac{-14(3 + \\sqrt{3})}{24}

We can now simplify the fraction by dividing both the numerator and the denominator by their common factor. The greatest common divisor (GCD) of 14 and 24 is 2. Dividing both by 2, we get:

frac−7(3+sqrt3)12\\frac{-7(3 + \\sqrt{3})}{12}

This is the simplified form of the expression. We have successfully rationalized the denominator and reduced the fraction to its simplest form. It's worth noting that simplification is not just about making the expression look neater; it often makes the expression easier to work with in further calculations. A simplified expression reduces the chances of errors in subsequent steps and can reveal underlying mathematical relationships more clearly. The process of simplification often involves a combination of factoring, canceling common factors, and applying algebraic identities. Mastering these techniques is crucial for success in algebra and beyond. With the expression now simplified to $\frac{-7(3 + \sqrt{3})}{12}$, we have completed the process of rationalizing the denominator and simplifying the original expression.

Final Result

After meticulously following the steps of identifying the conjugate, multiplying by the conjugate, and simplifying the expression, we have successfully rationalized the denominator of the original expression $\frac{-7}{6-2 \sqrt{3}}$. Our final simplified result is:

frac−7(3+sqrt3)12\\frac{-7(3 + \\sqrt{3})}{12}

This expression is equivalent to the original but has a rational denominator, making it easier to work with in various mathematical contexts. The process we've undertaken highlights the importance of several key mathematical concepts and techniques. We've utilized the concept of conjugates to eliminate the radical from the denominator, applied the difference of squares identity, and employed factorization and simplification to arrive at the final result. Each step was crucial, and a thorough understanding of the underlying principles is what allows us to confidently tackle such problems. It’s also important to recognize that there can be alternative forms of the final answer, depending on how the simplification is carried out. For instance, the expression could also be written as:

frac−21−7sqrt312\\frac{-21 - 7\\sqrt{3}}{12}

Both forms are mathematically equivalent, and the choice of which to use often depends on the specific context or the preference of the individual solving the problem. What’s crucial is the understanding of the process and the ability to manipulate the expression correctly. Rationalizing the denominator is not just a mechanical procedure; it’s a demonstration of mathematical reasoning and algebraic skill. It's a technique that is frequently used in higher-level mathematics, including calculus and complex analysis. Therefore, mastering this skill is an important step in your mathematical journey. In conclusion, the final simplified form of the expression $\frac{-7}{6-2 \sqrt{3}}$ with a rationalized denominator is $\frac{-7(3 + \sqrt{3})}{12}$, or equivalently, $\frac{-21 - 7\sqrt{3}}{12}$. This result showcases the power of algebraic manipulation in simplifying mathematical expressions.

Conclusion

In this comprehensive exploration, we have successfully rationalized the denominator and simplified the expression $\frac{-7}{6-2 \sqrt{3}}$. We began by understanding the concept of rationalizing the denominator and the reasons behind it. We then identified the conjugate of the denominator, which is a critical step in the process. By multiplying both the numerator and the denominator by this conjugate, we were able to eliminate the square root from the denominator. The subsequent step involved simplifying the resulting expression, which required factoring out common factors and reducing the fraction to its simplest form. Our journey through this problem highlights the interconnectedness of various mathematical concepts. We utilized the properties of conjugates, the difference of squares identity, and techniques for simplification. These are fundamental tools in algebra and are essential for tackling more complex mathematical problems. The final result, $\frac{-7(3 + \sqrt{3})}{12}$ or $\frac{-21 - 7\sqrt{3}}{12}$, demonstrates the effectiveness of rationalizing the denominator in simplifying expressions. It's important to remember that this process is not just about obtaining a numerical answer; it's about transforming an expression into a more manageable and understandable form. Rationalizing the denominator is a skill that extends far beyond this specific example. It is a crucial technique in various areas of mathematics, including calculus, complex analysis, and more. Mastering this skill will undoubtedly enhance your ability to solve a wide range of mathematical problems. As we conclude this exploration, it's worth emphasizing the importance of practice. The more you practice these techniques, the more comfortable and confident you will become in applying them. Mathematics is a discipline that rewards persistence and a deep understanding of fundamental principles. By mastering techniques like rationalizing the denominator, you are building a strong foundation for future mathematical endeavors.