Simplifying The Quotient Of Radicals A Mathematical Exploration

Embarking on a mathematical journey often involves deciphering complex expressions and simplifying them to their core essence. In this article, we delve into the realm of radicals and quotients, specifically focusing on the expression $(\sqrt{8} + \sqrt{11}) / (\sqrt{5} + \sqrt{3})$. Our mission is to unravel this expression and determine the correct simplified form from the options provided. This exploration will not only enhance our understanding of radical manipulation but also sharpen our problem-solving skills in mathematics.

The Challenge: Simplifying the Quotient of Radicals

The core of our task lies in simplifying the given quotient: $(\sqrt{8} + \sqrt{11}) / (\sqrt{5} + \sqrt{3})$. This expression involves radicals in both the numerator and the denominator, making it a prime candidate for a technique known as rationalizing the denominator. Rationalizing the denominator is a process that eliminates radicals from the denominator of a fraction, thereby making the expression simpler to handle and understand. This method is crucial when dealing with expressions that involve square roots or other radicals in the denominator, as it transforms the expression into a more manageable form. The options presented suggest that the simplified form will likely involve a combination of radicals and rational numbers, further emphasizing the need for a systematic approach to simplification.

Rationalizing the Denominator: A Step-by-Step Approach

To rationalize the denominator, we employ a clever strategy: multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(\sqrt{5} + \sqrt{3})$ is $(\sqrt{5} - \sqrt{3})$. This technique is based on the algebraic identity $(a + b)(a - b) = a^2 - b^2$, which eliminates the square roots in the denominator. By multiplying by the conjugate, we effectively transform the denominator into a rational number, paving the way for further simplification. This step is not just a mathematical trick; it's a fundamental technique in simplifying expressions involving radicals, allowing us to rewrite the expression in a more standard and easily understandable form.

Let's begin by multiplying both the numerator and the denominator by the conjugate:

$\frac{\sqrt{8} + \sqrt{11}}{\sqrt{5} + \sqrt{3}} \cdot \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}$

This multiplication sets the stage for expanding both the numerator and the denominator, which is the next crucial step in our simplification process. The careful expansion of these products will reveal how the radicals interact and how we can combine like terms to arrive at our final simplified expression. This process showcases the elegance of algebraic manipulation and its power in transforming complex expressions into simpler forms.

Expanding and Simplifying: Unveiling the Solution

Now, we expand the numerator and the denominator separately.

For the numerator, we have:

$(\sqrt{8} + \sqrt{11})(\sqrt{5} - \sqrt{3}) = \sqrt{8} \cdot \sqrt{5} - \sqrt{8} \cdot \sqrt{3} + \sqrt{11} \cdot \sqrt{5} - \sqrt{11} \cdot \sqrt{3}$

Simplifying this, we get:

$\sqrt{40} - \sqrt{24} + \sqrt{55} - \sqrt{33}$

We can further simplify $\sqrt{40}$ as $\sqrt{4 \cdot 10} = 2\sqrt{10}$ and $\sqrt{24}$ as $\sqrt{4 \cdot 6} = 2\sqrt{6}$. Thus, the numerator becomes:

$2\sqrt{10} - 2\sqrt{6} + \sqrt{55} - \sqrt{33}$

Next, we expand the denominator:

$(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = (\√5)^2 - (\sqrt{3})^2 = 5 - 3 = 2$

This elegant simplification in the denominator is the direct result of multiplying by the conjugate, which perfectly eliminates the radicals. Now we have a rational denominator, which makes the entire expression much easier to manage.

Putting It All Together: The Simplified Quotient

Combining the simplified numerator and denominator, we have:

$\frac{2\sqrt{10} - 2\sqrt{6} + \sqrt{55} - \sqrt{33}}{2}$

We can divide each term in the numerator by 2:

$\frac{2\sqrt{10}}{2} - \frac{2\sqrt{6}}{2} + \frac{\sqrt{55}}{2} - \frac{\sqrt{33}}{2} = \sqrt{10} - \sqrt{6} + \frac{\sqrt{55} - \sqrt{33}}{2}$

However, this form doesn't directly match any of the given options. Let's go back to the numerator before dividing by 2:

$\frac{2\sqrt{10} - 2\sqrt{6} + \sqrt{55} - \sqrt{33}}{2}$

We can rewrite $2\sqrt{10}$ as $\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$ and $2\sqrt{6}$ as $\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}$. Then we have:

$\frac{\sqrt{40} - \sqrt{24} + \sqrt{55} - \sqrt{33}}{2}$

Notice that $\sqrt{40}$ can be written as $\sqrt{5 \cdot 8} = \sqrt{5 \cdot 4 \cdot 2} = 2\sqrt{10}$ and $\sqrt{24}$ can be written as $\sqrt{3 \cdot 8} = \sqrt{3 \cdot 4 \cdot 2} = 2\sqrt{6}$. Rewriting these terms doesn't directly lead to a match with the given options either.

Let's revisit the numerator expansion: $\sqrt{40} - \sqrt{24} + \sqrt{55} - \sqrt{33}$. We can further break down the radicals:

$\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} = 2\sqrt{2 \cdot 5}$

$\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} = 2\sqrt{2 \cdot 3}$

Substituting these back into the numerator, we get:

$2\sqrt{10} - 2\sqrt{6} + \sqrt{55} - \sqrt{33}$

Let's examine the options provided more closely. Option B, $\frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}}{2}$, seems to be the closest match in terms of the radicals present. However, the term $-3\sqrt{2}$ is not immediately apparent in our simplified numerator. This suggests we might have missed a simplification step or that there's another way to express our radicals.

Let's go back to the original numerator expansion and try a different approach to simplifying the radicals:

$\sqrt{40} - \sqrt{24} + \sqrt{55} - \sqrt{33}$

$\sqrt{8 \cdot 5} - \sqrt{8 \cdot 3} + \sqrt{55} - \sqrt{33}$

$2\sqrt{2} \cdot \sqrt{5} - 2\sqrt{2} \cdot \sqrt{3} + \sqrt{55} - \sqrt{33}$

$2\sqrt{10} - 2\sqrt{6} + \sqrt{55} - \sqrt{33}$

This still doesn't directly match option B. However, let's consider option A, $\frac{\sqrt{30} + 3\sqrt{2} + \sqrt{55} + \sqrt{33}}{8}$. This option has a denominator of 8, which is different from our current denominator of 2. It's possible that there was an error in the problem statement or the options provided.

After careful re-evaluation and simplification, it appears there may be a discrepancy in the provided options. The closest simplified form we've obtained is:

$\frac{2\sqrt{10} - 2\sqrt{6} + \sqrt{55} - \sqrt{33}}{2}$

This doesn't directly correspond to any of the given choices. It's crucial to double-check the original problem and options for any potential errors. In mathematical problem-solving, accuracy is paramount, and sometimes the correct answer isn't among the options provided. This situation highlights the importance of critical thinking and the ability to recognize when a given set of answers might be flawed.

Conclusion: Navigating the Labyrinth of Radicals

In conclusion, simplifying the quotient $(\sqrt{8} + \sqrt{11}) / (\sqrt{5} + \sqrt{3})$ involved rationalizing the denominator and carefully expanding the resulting expressions. While we arrived at a simplified form, it didn't directly match any of the provided options, suggesting a potential error in the options themselves. This exercise underscores the importance of meticulous calculation and the ability to critically assess results in mathematical problem-solving. The journey through this problem has not only reinforced our understanding of radical manipulation but also highlighted the need for precision and attention to detail in mathematics. When faced with discrepancies, it's essential to revisit each step, ensuring accuracy and exploring alternative approaches, while also acknowledging the possibility of external errors in the problem statement or answer choices.