Simplifying The Quotient (2-√8)/(4+√12) A Step-by-Step Guide

In the realm of mathematics, simplifying complex expressions is a fundamental skill. Today, we embark on a journey to unravel the quotient (2-√8)/(4+√12). This seemingly intricate expression can be simplified through a series of algebraic manipulations and a keen understanding of radical properties. Our goal is to present a comprehensive explanation, ensuring clarity and fostering a deeper understanding of the underlying mathematical principles. This process involves rationalizing the denominator, simplifying radicals, and employing algebraic techniques to arrive at the most simplified form of the expression. Let's dive into the step-by-step solution and explore the elegance of mathematical simplification. We will break down each step, providing explanations and justifications to ensure that the process is transparent and easily followed. This detailed exploration will not only help in solving this particular problem but also equip you with the skills to tackle similar mathematical challenges with confidence. So, let's begin our mathematical journey and unravel the quotient (2-√8)/(4+√12) together.

The initial expression we are tasked with simplifying is (2-√8)/(4+√12). The presence of radicals in both the numerator and the denominator suggests that we need to rationalize the denominator to simplify this quotient. Rationalizing the denominator involves eliminating the radical from the denominator, which is typically achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. Before we proceed with this step, it's crucial to simplify the radicals present in the expression. Simplifying radicals makes the subsequent steps cleaner and easier to manage. Let's start by simplifying √8 and √12 individually. √8 can be expressed as √(42), which simplifies to 2√2. Similarly, √12 can be expressed as √(43), which simplifies to 2√3. By simplifying these radicals, we transform the original expression into (2-2√2)/(4+2√3). This simplified form is more manageable and sets the stage for the next step: rationalizing the denominator. Simplifying radicals at this stage is a crucial step in the simplification process, as it reduces the complexity of the numbers we are working with and makes the subsequent algebraic manipulations less cumbersome. Understanding and applying these initial simplification steps is key to efficiently solving mathematical expressions involving radicals.

To effectively tackle the quotient (2-√8)/(4+√12), our first step involves simplifying the radicals present in the expression. Radicals, particularly those with composite numbers under the root, can often be simplified into a more manageable form. This simplification not only makes the expression easier to work with but also reveals the underlying mathematical structure more clearly. Let's begin with √8. The number 8 can be factored into 4 multiplied by 2. Thus, √8 can be rewritten as √(42). The square root of a product is the product of the square roots, so √(42) becomes √4 * √2. Since √4 equals 2, we can simplify √8 to 2√2. Now, let's turn our attention to √12. The number 12 can be factored into 4 multiplied by 3. Therefore, √12 can be rewritten as √(43). Applying the same principle as before, √(43) becomes √4 * √3. As √4 equals 2, √12 simplifies to 2√3. By simplifying √8 to 2√2 and √12 to 2√3, we reduce the complexity of the original expression (2-√8)/(4+√12), transforming it into (2-2√2)/(4+2√3). This step is crucial as it lays the foundation for further simplification, particularly when it comes to rationalizing the denominator. Simplified radicals make the subsequent algebraic manipulations cleaner and less prone to errors. The ability to simplify radicals is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems.

Having simplified the radicals, our expression now stands as (2-2√2)/(4+2√3). The next pivotal step in simplifying this quotient is rationalizing the denominator. Rationalizing the denominator means eliminating the radical from the denominator, which in turn simplifies the overall expression. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression a + b is a – b. Therefore, the conjugate of our denominator, 4 + 2√3, is 4 – 2√3. Multiplying both the numerator and the denominator by 4 – 2√3, we get [(2-2√2) * (4-2√3)] / [(4+2√3) * (4-2√3)]. Let's focus on the denominator first. Multiplying (4+2√3) by (4-2√3) results in a difference of squares: (4)^2 - (2√3)^2, which simplifies to 16 - 12, and further to 4. Now, let's expand the numerator: (2-2√2) * (4-2√3) = 24 - 22√3 - 2√24 + 2√22√3 = 8 - 4√3 - 8√2 + 4√6. So, the expression now becomes (8 - 4√3 - 8√2 + 4√6) / 4. By rationalizing the denominator, we've eliminated the radical from the denominator, making the expression significantly simpler. The next step involves simplifying the resulting numerator and further reducing the fraction. Rationalizing the denominator is a standard technique in algebra and is crucial for simplifying expressions with radicals in the denominator. This process not only makes the expression easier to interpret but also facilitates further calculations and manipulations.

Following the rationalization of the denominator, our expression is now (8 - 4√3 - 8√2 + 4√6) / 4. The next step in our simplification journey is to further simplify this expression by factoring out common terms in the numerator and then reducing the fraction. We observe that each term in the numerator is divisible by 4. Factoring out 4 from the numerator, we get 4(2 - √3 - 2√2 + √6). Now, we can rewrite the expression as [4(2 - √3 - 2√2 + √6)] / 4. The 4 in the numerator and the 4 in the denominator cancel each other out, leaving us with the simplified numerator: 2 - √3 - 2√2 + √6. This simplified form is much cleaner and easier to work with. At this point, it's essential to examine whether further simplification is possible. We look for any like terms that can be combined or any further radical simplifications that can be made. In this case, there are no like terms to combine, and the radicals are in their simplest form. Therefore, the expression 2 - √3 - 2√2 + √6 represents the most simplified form of the numerator. By expanding the numerator and factoring out common terms, we've significantly reduced the complexity of the expression. This step demonstrates the power of algebraic manipulation in simplifying mathematical expressions. The ability to identify and factor out common terms is a crucial skill in algebra and is essential for simplifying a wide range of mathematical problems. This process not only makes the expression more concise but also reveals its underlying structure more clearly.

After rationalizing the denominator and simplifying the numerator, we have arrived at the expression 2 - √3 - 2√2 + √6. This form represents the final simplified version of the original quotient, (2-√8)/(4+√12). We have meticulously gone through each step, from simplifying radicals to rationalizing the denominator and expanding the numerator, to arrive at this solution. The journey of simplification has not only provided us with the answer but has also reinforced our understanding of the underlying mathematical principles. The final expression, 2 - √3 - 2√2 + √6, is in its most simplified form, as there are no further simplifications that can be made. There are no like terms to combine, and the radicals are in their simplest form. This result showcases the elegance of mathematical simplification, where a complex expression is transformed into a more concise and understandable form. By following a systematic approach and applying the appropriate algebraic techniques, we have successfully unraveled the quotient. The ability to simplify complex expressions is a crucial skill in mathematics, and this exercise has provided a valuable opportunity to hone this skill. The final simplified form not only provides the answer but also enhances our understanding of the mathematical relationships within the expression. The process of simplifying the quotient has highlighted the importance of each step, from simplifying radicals to rationalizing the denominator and expanding the numerator. This comprehensive approach ensures that we arrive at the most simplified form, which is both accurate and easy to interpret.

In conclusion, the simplification of the quotient (2-√8)/(4+√12) has been a rewarding mathematical journey. We began with a complex expression and, through a series of deliberate steps, arrived at its most simplified form: 2 - √3 - 2√2 + √6. This process underscored the importance of several key mathematical techniques, including simplifying radicals, rationalizing the denominator, and expanding and factoring expressions. Each step played a crucial role in transforming the original expression into its simplified form. The ability to simplify radicals allowed us to express √8 as 2√2 and √12 as 2√3, making the expression more manageable. Rationalizing the denominator, by multiplying both the numerator and the denominator by the conjugate of the denominator, eliminated the radical from the denominator and further simplified the expression. Expanding the numerator and factoring out common terms allowed us to reduce the fraction to its simplest form. The final result, 2 - √3 - 2√2 + √6, is a testament to the power of algebraic manipulation and the elegance of mathematical simplification. This exercise not only provided us with the solution but also reinforced our understanding of the underlying mathematical principles. The skills and techniques employed in this simplification process are applicable to a wide range of mathematical problems, making this a valuable learning experience. The journey from the initial complex quotient to the final simplified form highlights the importance of a systematic approach and attention to detail in mathematics. By mastering these techniques, we can confidently tackle similar mathematical challenges and appreciate the beauty of mathematical simplification.