Simplifying Radical Expressions A Step-by-Step Guide

This article delves into the simplification of complex expressions involving radicals, specifically focusing on the expression ($\frac{7+3\sqrt[]{3}}{2+\sqrt[]{3\ }) - {\frac{7-3\sqrt[]{3}}{2-\sqrt[]{3\ }}$). We aim to break down the steps involved in rationalizing the denominators, combining the terms, and arriving at the simplified form, which will be expressed as $a + b$. This exploration is crucial for anyone looking to enhance their algebraic manipulation skills and gain a deeper understanding of working with irrational numbers.

Understanding the Initial Expression

In this exploration of simplifying expressions with radicals, we begin by dissecting the given expression: $\frac{7+3\sqrt[]{3}}{2+\sqrt[]{3\ }} - \frac{7-3\sqrt[]{3}}{2-\sqrt[]{3\ }}$. This expression involves two fractions, each with a radical in the denominator. To simplify this, our primary goal is to eliminate the radicals from the denominators, a process known as rationalization. Rationalizing the denominator makes the expression easier to manipulate and combine terms. The core concept behind rationalization is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate is formed by changing the sign between the terms in the denominator. For example, the conjugate of $2 + \sqrt[]{3}$ is $2 - \sqrt[]{3}$, and vice versa. This approach leverages the difference of squares identity, $(a + b)(a - b) = a^2 - b^2$, which will help us eliminate the square root from the denominator. By applying this technique to both fractions, we can transform the expression into a more manageable form, paving the way for further simplification and combination of terms. Mastering this technique is essential for dealing with complex algebraic expressions and is a fundamental skill in algebra. The process not only simplifies the expression but also allows us to express it in a standard form, making it easier to compare and analyze. This foundational step sets the stage for the subsequent operations, ensuring that we can accurately determine the values of $a$ and $b$ in the simplified form $a + b$.

Rationalizing the Denominators

To effectively simplify expressions with radicals, the critical step is rationalizing the denominators of the fractions. We are given the expression $\frac{7+3\sqrt[]{3}}{2+\sqrt[]{3\ }} - \frac{7-3\sqrt[]{3}}{2-\sqrt[]{3\ }}$. The first fraction, $\frac{7+3\sqrt[]{3}}{2+\sqrt[]{3\ }}$, has a denominator of $2 + \sqrt[]{3}$. To rationalize this, we multiply both the numerator and the denominator by its conjugate, which is $2 - \sqrt[]{3}$. This gives us: $\frac{7+3\sqrt[]{3}}{2+\sqrt[]{3\ }} \times \frac{2-\sqrt[]{3}}{2-\sqrt[]{3\ }}$. Performing this multiplication, we get a new numerator of $(7 + 3\sqrt[]{3})(2 - \sqrt[]{3})$ and a new denominator of $(2 + \sqrt[]{3})(2 - \sqrt[]{3})$. Similarly, for the second fraction, $\frac{7-3\sqrt[]{3}}{2-\sqrt[]{3\ }}$, the denominator is $2 - \sqrt[]{3}$, and its conjugate is $2 + \sqrt[]{3}$. We multiply both the numerator and the denominator by $2 + \sqrt[]{3}$: $\frac{7-3\sqrt[]{3}}{2-\sqrt[]{3\ }} \times \frac{2+\sqrt[]{3}}{2+\sqrt[]{3\ }}$. This results in a numerator of $(7 - 3\sqrt[]{3})(2 + \sqrt[]{3})$ and a denominator of $(2 - \sqrt[]{3})(2 + \sqrt[]{3})$. The key to rationalizing the denominator lies in recognizing the difference of squares pattern in the denominator: $(a + b)(a - b) = a^2 - b^2$. This allows us to eliminate the square root, simplifying the expression significantly. By meticulously applying this technique to both fractions, we set the stage for further simplification and combination of like terms, ultimately leading to the expression in the desired $a + b$ form. This step is not just about eliminating the radical; it’s about transforming the expression into a format that is easier to work with and understand, a fundamental aspect of algebraic manipulation.

Expanding the Numerators and Denominators

As part of simplifying expressions with radicals, after rationalizing the denominators, we focus on expanding the numerators and denominators. For the first fraction, the multiplication in the numerator is $(7 + 3\sqrt[]{3})(2 - \sqrt[]{3})$. Expanding this gives us $7 \times 2 - 7\sqrt[]{3} + 3\sqrt[]{3} \times 2 - 3\sqrt[]{3} \times \sqrt[]{3} = 14 - 7\sqrt[]{3} + 6\sqrt[]{3} - 9$. Combining like terms, the numerator simplifies to $5 - \sqrt[]{3}$. For the denominator of the first fraction, we have $(2 + \sqrt[]{3})(2 - \sqrt[]{3})$. Using the difference of squares formula, this expands to $2^2 - (\sqrt[]{3})^2 = 4 - 3 = 1$. Similarly, for the second fraction, the numerator is $(7 - 3\sqrt[]{3})(2 + \sqrt[]{3})$. Expanding this gives us $7 \times 2 + 7\sqrt[]{3} - 3\sqrt[]{3} \times 2 - 3\sqrt[]{3} \times \sqrt[]{3} = 14 + 7\sqrt[]{3} - 6\sqrt[]{3} - 9$. Combining like terms, the numerator simplifies to $5 + \sqrt[]{3}$. The denominator of the second fraction is $(2 - \sqrt[]{3})(2 + \sqrt[]{3})$, which, as before, expands to $2^2 - (\sqrt[]{3})^2 = 4 - 3 = 1$. Now we have the simplified fractions $\frac{5 - \sqrt[]{3}}{1}$ and $\frac{5 + \sqrt[]{3}}{1}$. Expanding these expressions is a critical step because it allows us to combine the terms effectively. By carefully applying the distributive property and the difference of squares formula, we transform the expression into a form where we can easily subtract the two fractions. This process highlights the importance of accurate algebraic manipulation and attention to detail in simplifying complex expressions. The resulting simplified fractions are much easier to handle, setting the stage for the final steps in solving the problem.

Combining the Simplified Fractions

Having expanded and simplified the fractions, the next crucial step in simplifying expressions with radicals involves combining the simplified fractions. We have now arrived at the expression $\frac{5 - \sqrt[]{3}}{1} - \frac{5 + \sqrt[]{3}}{1}$. Since both fractions have a common denominator of 1, the subtraction becomes straightforward. We simply subtract the numerators: $(5 - \sqrt[]{3}) - (5 + \sqrt[]{3})$. Distributing the negative sign across the second term, we get $5 - \sqrt[]{3} - 5 - \sqrt[]{3}$. Now, we combine like terms: $5 - 5$ cancels out, and we are left with $-\sqrt[]{3} - \sqrt[]{3}$, which simplifies to $-2\sqrt[]{3}$. This resulting expression, $-2\sqrt[]{3}$, is in the form $a + b\sqrt[]{3}$, where $a$ is 0 and $b$ is -2. Combining the fractions is a pivotal step as it consolidates the individual simplified components into a single, coherent expression. The simplicity of this step, given the common denominator, underscores the effectiveness of the preceding steps of rationalization and expansion. It highlights how methodical algebraic manipulation can transform a complex-looking expression into a manageable and easily interpretable form. This process showcases the elegance and efficiency of mathematical techniques in simplifying problems. The final expression, $-2\sqrt[]{3}$, represents the simplified form of the original expression, ready for interpretation and further use if needed.

Identifying a and b

In the final stage of simplifying expressions with radicals, the focus shifts to identifying the values of $a$ and $b$. We have simplified the original expression to $-2\sqrt[]{3}$, and we want to express this in the form $a + b\sqrt[]{3}$. By comparing the simplified expression with the desired form, we can easily determine the values of $a$ and $b$. In this case, $-2\sqrt[]{3}$ can be written as $0 + (-2)\sqrt[]{3}$. Therefore, $a$ is 0, and $b$ is -2. This identification is a crucial step because it provides the final answer in the requested format. It demonstrates the culmination of all the previous steps, from rationalizing the denominators to combining the simplified fractions. The process of identifying $a$ and $b$ underscores the importance of understanding the structure of mathematical expressions and recognizing the components that make up the final result. This step also reinforces the concept that complex algebraic manipulations can lead to simple and elegant solutions when approached methodically. The clarity in the final answer, with $a = 0$ and $b = -2$, is a testament to the power of algebraic techniques in simplifying and representing mathematical expressions in a concise and meaningful way.

Therefore, $\frac{7+3\sqrt[]{3}}{2+\sqrt[]{3\ }} - \frac{7-3\sqrt[]{3}}{2-\sqrt[]{3\ }} = 0 + (-2)\sqrt[]{3}$.

Conclusion

In conclusion, the journey of simplifying expressions with radicals, specifically the expression $\frac{7+3\sqrt[]{3}}{2+\sqrt[]{3\ }} - \frac{7-3\sqrt[]{3}}{2-\sqrt[]{3\ }}$, has been a comprehensive exercise in algebraic manipulation. We began by understanding the initial expression and the need for rationalizing the denominators. We then meticulously applied the technique of multiplying both the numerator and the denominator of each fraction by the conjugate of its denominator. This crucial step eliminated the radicals from the denominators, paving the way for further simplification. Next, we expanded the numerators and denominators, carefully applying the distributive property and the difference of squares formula. This process transformed the expression into a form where the fractions could be easily combined. With the denominators rationalized and the numerators expanded, we combined the simplified fractions, leading to the expression $-2\sqrt[]{3}$. Finally, we identified the values of $a$ and $b$ by expressing the simplified expression in the form $a + b\sqrt[]{3}$, determining that $a = 0$ and $b = -2$. This exercise highlights the importance of a systematic approach to simplifying complex algebraic expressions. Each step, from rationalization to final identification, plays a vital role in arriving at the correct answer. The ability to manipulate radicals, apply algebraic identities, and combine like terms is fundamental to success in algebra and beyond. The simplified form, $0 + (-2)\sqrt[]{3}$, not only provides a clear and concise solution but also showcases the elegance and power of mathematical techniques in transforming and simplifying complex problems. This exploration underscores the value of mastering algebraic skills and the satisfaction of solving intricate mathematical challenges.