Simplifying (7√2) / (5 - 3√2) A Step-by-Step Solution

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In the realm of mathematics, simplifying expressions involving radicals and fractions often presents a fascinating challenge. This article delves into the process of finding the value of the expression 725−32{ \frac{7\sqrt{2}}{5 - 3\sqrt{2}} }. We will embark on a step-by-step journey, employing the technique of rationalizing the denominator to arrive at the simplified form. Along the way, we will explore the underlying mathematical principles and provide a clear, concise explanation for each step. This exploration will not only reveal the solution but also enhance your understanding of radical manipulation and algebraic simplification. Understanding these concepts is crucial for success in various mathematical domains, from algebra to calculus. By the end of this discussion, you will not only know the answer but also appreciate the methodology behind it. Let's embark on this mathematical adventure and unravel the value of this expression together. Our main focus will be on removing the radical from the denominator, a common practice in simplifying such expressions. This involves multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate is formed by changing the sign between the terms in the denominator. This process leverages the difference of squares identity, which helps eliminate the radical in the denominator. Through careful application of this technique and meticulous arithmetic, we will arrive at the simplified form of the expression, one of the options provided.

Step-by-Step Solution

1. Rationalizing the Denominator

The initial step in simplifying the expression 725−32{ \frac{7\sqrt{2}}{5 - 3\sqrt{2}} } is to rationalize the denominator. Rationalizing the denominator means eliminating the radical from the denominator of a fraction. This is achieved by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of 5−32{ 5 - 3\sqrt{2} } is 5+32{ 5 + 3\sqrt{2} }. This manipulation is based on the principle that multiplying a fraction by a form of 1 (in this case, 5+325+32{ \frac{5 + 3\sqrt{2}}{5 + 3\sqrt{2}} }) does not change its value. By multiplying by the conjugate, we set the stage for using the difference of squares identity, which will effectively eliminate the square root from the denominator. This step is crucial because it transforms the expression into a more manageable form, allowing us to simplify it further. Understanding the concept of conjugates and their role in rationalization is fundamental in algebra. It is a technique used extensively in simplifying expressions involving radicals, making it an essential skill for any mathematics student.

\begin{align*} \frac{7\sqrt{2}}{5 - 3\sqrt{2}} &= \frac{7\sqrt{2}}{5 - 3\sqrt{2}} \times \frac{5 + 3\sqrt{2}}{5 + 3\sqrt{2}} \ \end{align*}

2. Multiplying Numerator and Denominator

Now, we proceed with the multiplication in both the numerator and the denominator. In the numerator, we multiply 72{ 7\sqrt{2} } by (5+32){ (5 + 3\sqrt{2}) }, which requires distributing 72{ 7\sqrt{2} } across both terms inside the parentheses. In the denominator, we multiply (5−32){ (5 - 3\sqrt{2}) } by its conjugate (5+32){ (5 + 3\sqrt{2}) }. This is where the difference of squares identity comes into play. Recall that (a−b)(a+b)=a2−b2{ (a - b)(a + b) = a^2 - b^2 }. This identity allows us to eliminate the radical in the denominator. The process of multiplication is a fundamental algebraic operation, and its correct application is critical to obtaining the correct simplified expression. Careful attention to detail in this step is essential to avoid errors. The numerator multiplication involves distributing a term with a square root, while the denominator multiplication utilizes a special algebraic identity, showcasing the interplay between different mathematical concepts in solving a single problem. Mastering these multiplication techniques is crucial for success in algebra and beyond.

\begin{align*} &= \frac{7\sqrt{2} (5 + 3\sqrt{2})}{(5 - 3\sqrt{2})(5 + 3\sqrt{2})} \ &= \frac{7\sqrt{2} \cdot 5 + 7\sqrt{2} \cdot 3\sqrt{2}}{5^2 - (3\sqrt{2})^2} \ \end{align*}

3. Simplifying the Expression

In this step, we simplify the expressions obtained in the previous step. We begin by simplifying the numerator. The term 72⋅5{ 7\sqrt{2} \cdot 5 } simplifies to 352{ 35\sqrt{2} }. The term 72⋅32{ 7\sqrt{2} \cdot 3\sqrt{2} } simplifies to 21⋅2{ 21 \cdot 2 }, which equals 42. So, the numerator becomes 352+42{ 35\sqrt{2} + 42 }. Next, we simplify the denominator. 52{ 5^2 } equals 25, and (32)2{ (3\sqrt{2})^2 } equals 9⋅2{ 9 \cdot 2 }, which is 18. Therefore, the denominator simplifies to 25−18{ 25 - 18 }, which equals 7. This simplification process involves basic arithmetic operations combined with the properties of square roots. Attention to detail and careful calculation are essential to ensure accuracy in this step. Simplifying expressions is a fundamental skill in mathematics, and it often involves breaking down complex expressions into simpler, more manageable terms. The ability to simplify expressions efficiently is crucial for problem-solving in various mathematical contexts.

\begin{align*} &= \frac{35\sqrt{2} + 42}{25 - 18} \ &= \frac{35\sqrt{2} + 42}{7} \ \end{align*}

4. Final Simplification

Finally, we perform the last step of simplification. We observe that both terms in the numerator, 352{ 35\sqrt{2} } and 42, are divisible by 7. We can factor out 7 from the numerator, which gives us 7(52+6){ 7(5\sqrt{2} + 6) }. The expression then becomes 7(52+6)7{ \frac{7(5\sqrt{2} + 6)}{7} }. Now, we can cancel the 7 in the numerator and the denominator, leaving us with the simplified expression 52+6{ 5\sqrt{2} + 6 }, which can also be written as 6+52{ 6 + 5\sqrt{2} }. This final step demonstrates the power of factorization and cancellation in simplifying mathematical expressions. Recognizing common factors and being able to simplify fractions are crucial skills in algebra. This step also highlights the importance of expressing the final answer in its simplest form, which is a common expectation in mathematical problem-solving. The ability to manipulate and simplify expressions like this is essential for success in more advanced mathematical topics.

\begin{align*} &= \frac{7(5\sqrt{2} + 6)}{7} \ &= 6 + 5\sqrt{2} \ \end{align*}

Conclusion

Therefore, the value of 725−32{ \frac{7\sqrt{2}}{5 - 3\sqrt{2}} } is 6+52{ 6 + 5\sqrt{2} }, which corresponds to option (A). This problem highlights the importance of rationalizing the denominator when dealing with expressions involving radicals in the denominator. The process involves multiplying both the numerator and the denominator by the conjugate of the denominator, which eliminates the radical from the denominator. The solution also reinforces the significance of careful algebraic manipulation and simplification. Each step, from rationalizing the denominator to the final simplification, requires a clear understanding of algebraic principles and meticulous execution. Mastering these skills is essential for anyone pursuing mathematics or related fields. The ability to simplify expressions, work with radicals, and apply algebraic identities is fundamental to success in more advanced mathematical topics. This problem serves as a valuable exercise in reinforcing these essential mathematical skills.

Simplifying Radical Expressions The Value of 7√2 / (5 - 3√2)