Rationalize Denominator Express In The Form A√21+b√6

In the realm of mathematics, particularly when dealing with surds (expressions involving square roots or other radicals), rationalizing the denominator is a fundamental technique. This process eliminates radicals from the denominator of a fraction, making it easier to work with and compare different expressions. In this comprehensive guide, we will delve into the intricacies of rationalizing denominators, focusing on expressions of the form $\frac{10 \sqrt{3}}{\sqrt{7}-\sqrt{2}}$. This specific example not only illustrates the core principles but also showcases the practical application of algebraic manipulation to achieve a simplified form, specifically $a \sqrt{21}+b \sqrt{6}$, where 'a' and 'b' are rational numbers. Understanding how to rationalize denominators is essential for simplifying complex expressions, solving equations, and performing various mathematical operations with greater ease and accuracy. This skill is particularly useful in algebra, calculus, and other advanced mathematical fields. The ability to manipulate surds effectively is a hallmark of strong mathematical proficiency, allowing for clearer insights and more efficient problem-solving strategies.

Before diving into the specifics of our example, it's important to establish a firm grasp on the underlying concepts. Surds, or radicals, are expressions that involve the root of a number, most commonly square roots. A surd in the denominator of a fraction can make it difficult to perform arithmetic operations or compare values. Rationalizing the denominator is the process of transforming the fraction into an equivalent form where the denominator is a rational number (i.e., it does not contain any radicals). This is typically achieved by multiplying both the numerator and the denominator by a carefully chosen expression, often the conjugate of the denominator. The conjugate of a binomial expression involving surds, such as $\sqrt{7}-\sqrt{2}$, is formed by changing the sign between the terms, resulting in $\sqrt{7}+\sqrt{2}$. When a binomial expression is multiplied by its conjugate, the radical terms cancel out due to the difference of squares identity, $(a - b)(a + b) = a^2 - b^2$. This fundamental algebraic principle is the key to rationalizing denominators effectively. The process not only simplifies expressions but also enhances our understanding of algebraic manipulations and their practical applications in various mathematical contexts. Mastering this technique is a crucial step in developing a strong foundation in algebra and related mathematical disciplines.

Now, let's apply the principle of rationalizing the denominator to our specific expression: $\frac{10 \sqrt{3}}{\sqrt{7}-\sqrt{2}}$. The first step is to identify the conjugate of the denominator. As we discussed, the conjugate of $\sqrt{7}-\sqrt{2}$ is $\sqrt{7}+\sqrt{2}$. We then multiply both the numerator and the denominator of the fraction by this conjugate. This ensures that we are multiplying the fraction by 1, thereby preserving its value while changing its form. The multiplication yields:

$$\frac{10 \sqrt{3}}{\sqrt{7}-\sqrt{2}} \times \frac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}}$$

Next, we perform the multiplication in both the numerator and the denominator. In the numerator, we distribute $10 \sqrt{3}$ across the terms $\sqrt{7}$ and $\sqrt{2}$. In the denominator, we apply the difference of squares identity. The numerator becomes:

$$10 \sqrt{3}(\sqrt{7}+\sqrt{2}) = 10 \sqrt{21} + 10 \sqrt{6}$$

The denominator, using the difference of squares, simplifies to:

$$(\sqrt{7}-\sqrt{2})(\sqrt{7}+\sqrt{2}) = (\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5$$

Thus, our expression now looks like:

$$\frac{10 \sqrt{21} + 10 \sqrt{6}}{5}$$

Finally, we simplify the fraction by dividing both terms in the numerator by the denominator:

$$\frac{10 \sqrt{21} + 10 \sqrt{6}}{5} = 2 \sqrt{21} + 2 \sqrt{6}$$

This result matches the desired form $a \sqrt{21}+b \sqrt{6}$, where $a = 2$ and $b = 2$. The step-by-step process highlights the importance of identifying the conjugate, applying the distributive property, and simplifying the resulting expression to achieve the rationalized form. This methodical approach ensures accuracy and clarity in manipulating surd expressions.

To further clarify the process, let's break down each step with meticulous detail. The initial expression is $\frac{10 \sqrt{3}}{\sqrt{7}-\sqrt{2}}$. As we've established, the key to rationalizing the denominator lies in multiplying both the numerator and denominator by the conjugate of the denominator, which in this case is $\sqrt{7}+\sqrt{2}$. This operation can be represented as:

$$\frac{10 \sqrt{3}}{\sqrt{7}-\sqrt{2}} \times \frac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}}$$

Now, let's focus on the numerator. We need to distribute $10 \sqrt{3}$ across the terms $(\sqrt{7}+\sqrt{2})$. This involves multiplying $10 \sqrt{3}$ by both $\sqrt{7}$ and $\sqrt{2}$ separately. The multiplication yields:

$$10 \sqrt{3} \times \sqrt{7} = 10 \sqrt{3 \times 7} = 10 \sqrt{21}$$

And,

$$10 \sqrt{3} \times \sqrt{2} = 10 \sqrt{3 \times 2} = 10 \sqrt{6}$$

Combining these results, the numerator becomes:

$$10 \sqrt{21} + 10 \sqrt{6}$$

Moving on to the denominator, we apply the difference of squares identity. Multiplying $(\sqrt{7}-\sqrt{2})$ by $(\sqrt{7}+\sqrt{2})$ gives:

$$(\sqrt{7}-\sqrt{2})(\sqrt{7}+\sqrt{2}) = (\sqrt{7})^2 - (\sqrt{2})^2$$

Squaring the square roots simplifies to:

$$7 - 2 = 5$$

Therefore, the denominator simplifies to 5. Now, we have the expression:

$$\frac{10 \sqrt{21} + 10 \sqrt{6}}{5}$$

To complete the rationalization, we divide each term in the numerator by the denominator:

$$\frac{10 \sqrt{21}}{5} + \frac{10 \sqrt{6}}{5}$$

This simplifies to:

$$2 \sqrt{21} + 2 \sqrt{6}$$

Thus, the final rationalized form is $2 \sqrt{21} + 2 \sqrt{6}$, which confirms that $a = 2$ and $b = 2$. This detailed breakdown reinforces the importance of each step in the process, from identifying the conjugate to simplifying the resulting expression. By carefully executing each step, we can confidently rationalize denominators and manipulate surd expressions with precision.

Rationalizing denominators, while a fundamental technique, can be prone to errors if not approached carefully. Understanding common mistakes is crucial for ensuring accuracy and mastering the process. One frequent error is incorrectly identifying the conjugate of the denominator. For a binomial expression like $\sqrt{7}-\sqrt{2}$, the conjugate is $\sqrt{7}+\sqrt{2}$. A common mistake is to simply change the sign of one term, for example, using $-\sqrt{7}-\sqrt{2}$ as the conjugate. This will not lead to the elimination of radicals in the denominator. To avoid this, always remember that the conjugate is formed by changing the sign between the two terms of the binomial. Another common mistake occurs during the multiplication process. When distributing the numerator, it's essential to correctly multiply the terms involving surds. For example, when multiplying $10 \sqrt{3}$ by $(\sqrt{7}+\sqrt{2})$, each term must be multiplied separately: $10 \sqrt{3} \times \sqrt{7} = 10 \sqrt{21}$ and $10 \sqrt{3} \times \sqrt{2} = 10 \sqrt{6}$. Errors can arise if the multiplication is not performed meticulously. Similarly, in the denominator, the difference of squares identity must be applied correctly. For $(\sqrt{7}-\sqrt{2})(\sqrt{7}+\sqrt{2})$, the result should be $(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5$. A mistake here could involve incorrect squaring of the surds or misapplication of the difference of squares formula. Simplification is another area where errors can occur. After rationalizing, the expression may need further simplification. For instance, if the numerator and denominator have a common factor, it should be divided out to obtain the simplest form. In our example, $\frac{10 \sqrt{21} + 10 \sqrt{6}}{5}$ simplifies to $2 \sqrt{21} + 2 \sqrt{6}$. Failing to simplify can leave the answer incomplete. To avoid these pitfalls, a systematic approach is key. Double-check the conjugate, perform multiplications carefully, apply algebraic identities accurately, and always simplify the final result. Regular practice and attention to detail will significantly reduce the likelihood of errors and solidify your understanding of rationalizing denominators. By being aware of these common mistakes and actively working to avoid them, you can enhance your proficiency in algebraic manipulation and problem-solving.

While rationalizing denominators might seem like a purely theoretical exercise, it has significant practical applications in various fields of mathematics, science, and engineering. In mathematics, rationalizing denominators is crucial for simplifying expressions and making them easier to work with. It allows for straightforward comparisons of different expressions and facilitates further calculations. For example, when solving equations involving surds, rationalizing the denominator can transform a complex equation into a more manageable form. In calculus, rationalizing the denominator is often a necessary step in evaluating limits and derivatives. Expressions with surds in the denominator can be difficult to differentiate or integrate directly, but rationalizing the denominator can simplify the process significantly. In physics and engineering, surds frequently appear in calculations involving distances, forces, and energies. Rationalizing the denominator can lead to more accurate and easily interpretable results. For instance, in electrical engineering, impedance calculations often involve complex numbers with surds in the denominator. Rationalizing the denominator makes it easier to determine the magnitude and phase of the impedance. Similarly, in mechanics, calculations involving projectile motion or simple harmonic motion may involve surds. Rationalizing the denominator simplifies these calculations and provides clearer insights into the physical phenomena. Furthermore, in computer graphics and image processing, surds are used in calculations related to transformations and projections. Rationalizing the denominator can improve the efficiency and accuracy of these computations. The importance of rationalizing denominators extends beyond specific applications. It is a fundamental skill that enhances mathematical fluency and problem-solving abilities. Mastering this technique provides a solid foundation for tackling more advanced mathematical concepts and real-world problems. The ability to manipulate surds and simplify expressions is a valuable asset in any field that relies on mathematical reasoning and computation. By understanding the principles and practicing the techniques of rationalizing denominators, students and professionals alike can improve their analytical skills and achieve greater success in their respective domains. In conclusion, rationalizing denominators is not just a mathematical trick; it is a powerful tool with wide-ranging applications that underpins many aspects of scientific and engineering practice.

In this comprehensive exploration, we have thoroughly examined the process of rationalizing the denominator, specifically focusing on the expression $\frac{10 \sqrt{3}}{\sqrt{7}-\sqrt{2}}$. We have dissected each step, from identifying the conjugate to simplifying the final result, ensuring a clear and methodical understanding of the technique. The final rationalized form, $2 \sqrt{21} + 2 \sqrt{6}$, exemplifies the power of algebraic manipulation in transforming complex expressions into simpler, more manageable forms. This skill is not only valuable in academic settings but also essential for various real-world applications in mathematics, science, and engineering. Mastering the art of rationalizing denominators involves more than just memorizing steps; it requires a deep understanding of the underlying principles and a keen eye for detail. By recognizing common mistakes and actively working to avoid them, you can enhance your accuracy and efficiency in manipulating surd expressions. The detailed breakdown of each calculation, as provided in this guide, serves as a valuable resource for reinforcing your understanding and honing your problem-solving skills. The importance of this technique extends beyond the immediate task of simplifying fractions. It fosters a broader appreciation for algebraic manipulation and its role in making complex calculations more accessible. Rationalizing denominators is a building block for more advanced mathematical concepts and a testament to the elegance and utility of mathematical tools. As you continue your mathematical journey, the skills and insights gained from this exploration will undoubtedly prove invaluable. The ability to confidently and accurately rationalize denominators is a hallmark of mathematical proficiency and a key to unlocking further mathematical challenges. Therefore, take the time to practice and solidify your understanding, and you will find yourself well-equipped to tackle a wide range of problems involving surds and radicals. The journey to mathematical mastery is paved with fundamental techniques like these, and each step forward brings you closer to a deeper appreciation of the power and beauty of mathematics.