Simplifying (8 - 5√6) / (√3 + √2) + 23√2 A Step-by-Step Solution

In this article, we will delve into the process of simplifying the expression (8 - 5√6) / (√3 + √2) + 23√2. This mathematical problem combines concepts of algebraic manipulation, rationalizing denominators, and simplifying radicals. We will break down each step in detail to ensure a clear understanding of the solution. Let's embark on this mathematical journey to uncover the simplified form of the given expression. This exploration is essential for anyone looking to enhance their algebra skills, particularly in handling expressions involving radicals. The content is structured to guide you through each phase of the simplification process, providing insights and techniques that can be applied to similar problems.

Understanding the Initial Expression

The given expression is (8 - 5√6) / (√3 + √2) + 23√2. To simplify this, our primary goal is to eliminate the radical in the denominator of the fraction. This process is known as rationalizing the denominator. To achieve this, we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of √3 + √2 is √3 - √2. Understanding this initial step is crucial as it sets the stage for subsequent simplifications. The entire problem revolves around the manipulation of radicals and rational numbers, a fundamental aspect of algebraic simplification. By understanding the structure of the expression, we can apply the appropriate techniques to reach a simplified form.

Rationalizing the Denominator

The crucial step here involves multiplying both the numerator and the denominator of the fractional part of the expression by the conjugate of the denominator. This technique eliminates the radicals from the denominator. We multiply (8 - 5√6) by (√3 - √2) and (√3 + √2) by (√3 - √2). This process is rooted in the algebraic identity (a + b)(a - b) = a² - b², which helps to remove the square roots from the denominator. The multiplication in the numerator will involve distribution, a key algebraic skill. The denominator, after multiplication, will simplify to a rational number, which is our goal. This step is pivotal, as it transforms the initial complex fraction into a more manageable form, setting the stage for further simplification. Rationalizing the denominator is a common technique in simplifying expressions with radicals and is a fundamental skill in algebra.

Multiplying the Numerator

Now, let's focus on multiplying the numerator: (8 - 5√6)(√3 - √2). This involves distributing each term in the first parenthesis across the terms in the second parenthesis. We will multiply 8 by both √3 and -√2, and then we'll multiply -5√6 by both √3 and -√2. This process will result in four terms, each of which needs to be carefully simplified. The multiplication of the radical terms will involve using the property √a * √b = √(ab)*. It is important to keep track of the signs and coefficients during this multiplication process to avoid errors. This step is a classic example of algebraic expansion, a technique used extensively in simplifying expressions. The result of this multiplication will form the new numerator of our fraction.

Multiplying the Denominator

Next, we will multiply the denominator: (√3 + √2)(√3 - √2). As mentioned earlier, this multiplication is based on the algebraic identity (a + b)(a - b) = a² - b². Applying this identity, we get (√3)² - (√2)². Squaring a square root simply gives us the number under the radical. So, (√3)² is 3 and (√2)² is 2. The subtraction then gives us the simplified denominator. This step is a straightforward application of a well-known algebraic identity, making the process efficient and clear. The result is a rational number, which is the desired outcome of rationalizing the denominator. This simplification is crucial as it allows us to proceed with further steps in solving the original expression.

Simplifying the Expression

After multiplying the numerator and the denominator, we simplify the resulting expression. The numerator, after expansion and simplification, will contain terms with radicals, while the denominator will be a rational number. At this stage, we will combine like terms in the numerator, which includes terms with the same radical. The simplified numerator will then be divided by the simplified denominator. This step involves careful arithmetic and algebraic manipulation. We will also look for opportunities to further simplify the radicals, if any. This process of simplification is essential for arriving at the most concise form of the expression. The goal is to reduce the complexity of the expression while maintaining its mathematical equivalence.

Combining Like Terms

In this phase, we focus on identifying and combining like terms in the numerator. Like terms are those that have the same radical part. For example, terms with √2 can be combined, and terms with √3 can be combined. This combination involves adding or subtracting the coefficients of the like terms. It's important to ensure that only like terms are combined, as combining unlike terms would be mathematically incorrect. This step is a standard practice in algebraic simplification and is crucial for reducing the expression to its simplest form. The careful combination of like terms is a key factor in arriving at the final simplified answer.

Dividing and Simplifying Radicals

Once we have combined like terms in the numerator, we divide the entire numerator by the simplified denominator. This division might result in the simplification of coefficients. Additionally, we will look for opportunities to simplify the radicals themselves. This can involve factoring out perfect squares from under the radical sign. For example, √12 can be simplified to 2√3 because 12 can be factored as 4 * 3, and 4 is a perfect square. Simplifying radicals is an essential part of the overall simplification process. This step ensures that the expression is in its most reduced and understandable form. The process may involve several steps, but the goal is to express the radicals in their simplest form.

Adding the Remaining Term

Now, we bring back the remaining term, 23√2, and add it to the simplified expression we obtained from the fractional part. This step involves combining like terms once again. We will add the coefficient of √2 in the simplified expression to 23. This is a straightforward addition, but it's important to ensure that we are only adding like terms. The result of this addition will give us the final coefficient of √2 in the simplified expression. This step is the culmination of all the previous simplifications and brings us closer to the final answer. The final expression should be as concise and simplified as possible.

Final Simplified Form

After performing all the simplifications and combining like terms, we arrive at the final simplified form of the expression. This form will be the most concise representation of the original expression. It will typically involve integer coefficients and simplified radicals. The final answer should be checked for accuracy, ensuring that all steps were performed correctly. The process of simplification not only leads to a final answer but also enhances our understanding of algebraic manipulations and radical expressions. This simplified form is the ultimate result of our step-by-step simplification process.

Conclusion

In conclusion, the expression (8 - 5√6) / (√3 + √2) + 23√2 can be simplified through a series of algebraic manipulations, including rationalizing the denominator, expanding expressions, combining like terms, and simplifying radicals. The process involves careful attention to detail and a solid understanding of algebraic principles. By following each step methodically, we can arrive at the final simplified form of the expression. This exercise demonstrates the power of algebraic techniques in simplifying complex mathematical expressions. The ability to simplify such expressions is a valuable skill in mathematics and related fields.