Simplifying $\sqrt{35 W^5 X^6} \sqrt{7 W^2 X^5}$ A Step-by-Step Guide
In the realm of mathematics, simplifying expressions involving square roots is a fundamental skill. This guide provides a comprehensive, step-by-step approach to simplify the expression $\sqrt{35 w^5 x^6} \sqrt{7 w^2 x^5}$, assuming all variables represent positive real numbers. We will delve into the properties of square roots and exponents, demonstrating how to effectively manipulate and combine terms to arrive at the simplest form. This exploration is not just an exercise in algebraic manipulation; it's an opportunity to deepen our understanding of mathematical principles and enhance our problem-solving abilities. Mastery of these techniques is crucial for success in more advanced mathematical studies, making this a valuable skill for students and enthusiasts alike.
Understanding the Fundamentals of Square Roots
Before we dive into the simplification process, it's crucial to grasp the fundamental properties of square roots. The square root of a number a, denoted as $\sqrt{a}$, is a value that, when multiplied by itself, equals a. For instance, the square root of 9 is 3 because 3 * 3 = 9. When dealing with variables and exponents within square roots, we need to remember that $\sqrt{a^2} = |a|$, but since we are assuming all variables are positive, we can simplify this to $\sqrt{a^2} = a$. This concept is vital when simplifying expressions involving variables raised to powers. Another key property is the product rule of square roots, which states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This rule allows us to combine separate square roots into a single square root, which is essential for simplifying the given expression. Understanding these basic principles will pave the way for a smooth and efficient simplification process. The ability to apply these rules accurately and confidently is the cornerstone of success in simplifying more complex expressions. By mastering these fundamentals, we build a strong foundation for tackling various algebraic challenges. Furthermore, this foundational knowledge enhances our ability to recognize patterns and apply appropriate simplification techniques across different mathematical contexts. Therefore, a thorough understanding of square root properties is not just beneficial for this specific problem, but it also serves as a valuable asset in broader mathematical endeavors.
Step 1: Combining the Square Roots
The first step in simplifying the expression $\sqrt{35 w^5 x^6} \sqrt{7 w^2 x^5}$ is to combine the two square roots into a single square root using the product rule mentioned earlier. This involves multiplying the terms inside the square roots together. So, we have:
This step is crucial because it consolidates the expression, making it easier to manage and simplify. By bringing all the terms under one square root, we can then apply other simplification rules more effectively. The multiplication within the square root sets the stage for combining like terms and reducing the expression to its simplest form. This initial combination is not just a procedural step; it's a strategic move that streamlines the simplification process. It allows us to see the expression as a whole, rather than dealing with separate parts, which is essential for identifying common factors and applying the rules of exponents and square roots. Moreover, this step highlights the importance of understanding and applying the fundamental properties of square roots, demonstrating how these properties can be used to simplify complex expressions. By mastering this initial step, we lay the groundwork for a more efficient and accurate simplification process.
Step 2: Multiplying the Terms Inside the Square Root
Now that we have combined the square roots, the next step is to multiply the terms inside the square root. This involves multiplying the coefficients (the numerical parts) and then multiplying the variables with their respective exponents. Recall that when multiplying variables with exponents, we add the exponents together (e.g., $w^5 \cdot w^2 = w^{5+2} = w^7$). Applying this to our expression, we get:
Multiplying the coefficients, 35 and 7, gives us 245. Multiplying the variables, we add the exponents: $w^5 \cdot w^2 = w^7$ and $x^6 \cdot x^5 = x^{11}$. So, the expression inside the square root becomes:
This step is critical for organizing the terms and preparing them for further simplification. By performing the multiplication, we consolidate the expression into a single term under the square root, making it easier to identify perfect square factors and apply the properties of exponents. This process not only simplifies the expression but also enhances our understanding of how different mathematical operations interact within a single problem. The careful execution of this step ensures that we are on the right track towards the final simplified form. Furthermore, it reinforces the importance of precision and attention to detail in mathematical manipulations, highlighting how a small error in one step can propagate through the entire solution process.
Step 3: Factoring and Simplifying the Square Root
The next crucial step involves factoring the coefficient and simplifying the square root. To simplify $\sqrt{245 w^7 x^{11}}$, we need to identify any perfect square factors within 245 and rewrite the variables with even exponents, as these can be easily extracted from the square root. Let's start by factoring 245. The prime factorization of 245 is $5 \cdot 7 \cdot 7$, which can be written as $5 \cdot 7^2$. We can see that 7 squared is a perfect square factor.
Now, let's address the variables. For $w^7$, we can rewrite it as $w^6 \cdot w$, where $w^6$ is a perfect square (since 6 is an even number). Similarly, for $x^{11}$, we can rewrite it as $x^{10} \cdot x$, where $x^{10}$ is also a perfect square. Rewriting the expression, we get:
This step is pivotal in isolating the perfect square factors, which will allow us to simplify the square root. By identifying and separating these factors, we can extract their square roots and move them outside the radical. This process demonstrates a deep understanding of the properties of square roots and exponents, as well as the ability to manipulate expressions strategically. The careful factoring and rewriting of terms is essential for achieving the simplest form of the expression. Moreover, this step highlights the importance of number sense and algebraic fluency in simplifying mathematical expressions, showcasing how a combination of these skills can lead to efficient and accurate solutions. By mastering this technique, we can confidently tackle a wide range of simplification problems involving square roots and exponents.
Step 4: Extracting the Square Roots
With the expression factored into perfect squares and remaining terms, we can now extract the square roots of the perfect square factors. Recall that $\sqrt{a^2} = a$, $\sqrt{w^6} = w^3$, and $\sqrt{x^{10}} = x^5$. Applying this to our expression:
Taking the square roots of the perfect squares, we get:
This step is where the simplification truly comes to fruition. By extracting the square roots of the perfect square factors, we significantly reduce the complexity of the expression. This process not only simplifies the mathematical form but also provides a clearer understanding of the relationship between the terms. The ability to confidently and accurately extract these square roots is a testament to a solid grasp of the fundamental properties of square roots and exponents. Moreover, this step showcases the power of strategic factoring in simplifying expressions, demonstrating how breaking down a complex problem into smaller, manageable parts can lead to an elegant solution. The resulting expression is now in its simplest form, with all possible square roots extracted, highlighting the effectiveness of the step-by-step approach we have followed.
Final Answer
Therefore, the simplified form of the expression $\sqrt{35 w^5 x^6} \sqrt{7 w^2 x^5}$ is:
This final answer represents the culmination of our step-by-step simplification process. It is the most concise and simplified form of the original expression, achieved by systematically applying the properties of square roots and exponents. The journey from the initial complex expression to this simplified form demonstrates the power of mathematical manipulation and the importance of understanding fundamental principles. This result is not just a numerical or algebraic solution; it's a testament to our ability to analyze, strategize, and execute mathematical operations with precision and clarity. The process we've undertaken highlights the beauty of mathematics in its ability to transform complex problems into elegant solutions. Furthermore, this final answer serves as a tangible representation of the skills and knowledge gained throughout the simplification process, reinforcing the value of mathematical education and its ability to empower us to solve challenging problems.
Conclusion
In conclusion, simplifying expressions involving square roots requires a thorough understanding of the properties of square roots and exponents, as well as a systematic approach. By combining the square roots, multiplying the terms, factoring, extracting square roots, and presenting the final answer, we have successfully simplified the given expression. This process not only provides the simplified form but also enhances our mathematical skills and problem-solving abilities. The ability to simplify complex expressions is a valuable asset in various mathematical contexts, and mastering these techniques is crucial for success in advanced studies. The systematic approach we've employed serves as a model for tackling other mathematical challenges, emphasizing the importance of breaking down complex problems into smaller, manageable steps. Moreover, this journey through simplification highlights the interconnectedness of mathematical concepts and the power of applying fundamental principles to achieve elegant solutions. The skills and insights gained from this exercise extend beyond the specific problem, empowering us to approach future mathematical challenges with confidence and competence. Therefore, mastering the art of simplifying expressions is not just about finding the answer; it's about developing a deeper understanding of mathematical principles and enhancing our overall problem-solving capabilities.
