Simplify Radical Expressions With Variables A Step By Step Guide

In the realm of mathematics, simplifying radical expressions is a fundamental skill, especially when dealing with variables within those expressions. This article delves into the process of combining radical terms using indicated operations, assuming all variables are positive. We will dissect the given expression: $7 x \sqrt{775 x y}+8 x \sqrt{31 x y}-8 x \sqrt{775 x y}$, providing a step-by-step guide to simplification. Our primary focus is to ensure clarity and comprehension, making this guide accessible to learners of all levels. Understanding how to simplify radical expressions is crucial not only for academic success but also for various practical applications in fields like physics, engineering, and computer science. By mastering these techniques, you'll be able to tackle more complex mathematical problems with confidence and precision.

When dealing with radical expressions, the first step in simplification often involves identifying and combining like terms. Like terms are those that have the same radical part. In our expression, we have three terms: $7 x \sqrt{775 x y}$, $8 x \sqrt{31 x y}$, and $-8 x \sqrt{775 x y}$. Notice that the first and third terms have the same radical component, $\sqrt{775 x y}$. This means they can be combined directly. Combining like terms is similar to combining variables in algebraic expressions; you simply add or subtract the coefficients of the terms that share the same radical. Think of the radical part as a common unit, like 'x' in the expression 7x + 8y - 8x. Just as you would combine 7x and -8x, you can combine terms with the same radical. This initial step is crucial because it reduces the number of terms and often simplifies the expression significantly. By focusing on identifying and combining like radicals, you set the stage for further simplification and make the overall process more manageable. Remember, the key is to treat the radical part as a single entity when combining terms, which is a core concept in simplifying radical expressions.

The next crucial step in simplifying radical expressions is to look for opportunities to simplify the radicals themselves. This often involves factoring the radicand (the expression inside the square root) and looking for perfect square factors. In our expression, we have radicals like $\sqrt{775 x y}$ and $\sqrt{31 x y}$. To simplify $\sqrt{775 x y}$, we first need to factor 775. The prime factorization of 775 is $5^2 \cdot 31$. This means we can rewrite $\sqrt{775 x y}$ as $\sqrt{5^2 \cdot 31 x y}$. The presence of $5^2$ (a perfect square) allows us to simplify the radical further. We can take the square root of $5^2$, which is 5, and move it outside the radical. This transforms $\sqrt{5^2 \cdot 31 x y}$ into $5 \sqrt{31 x y}$. Now, let's consider the second radical, $\sqrt{31 x y}$. Since 31 is a prime number and x and y are variables, this radical cannot be simplified further. Identifying perfect square factors within radicals is a key technique in simplification, as it allows you to reduce the radicand to its simplest form. This not only makes the expression easier to work with but also provides a clearer understanding of the terms involved. By mastering this factoring technique, you'll be well-equipped to simplify a wide variety of radical expressions.

Let's embark on a step-by-step simplification of the given expression: $7 x \sqrt{775 x y}+8 x \sqrt{31 x y}-8 x \sqrt{775 x y}$. This process will demonstrate the practical application of the concepts discussed earlier. The first step is to identify and combine like terms. As we noted before, the terms $7 x \sqrt{775 x y}$ and $-8 x \sqrt{775 x y}$ share the same radical part, $\sqrt{775 x y}$. We can combine these terms by adding their coefficients: $7x - 8x = -1x$. So, combining these terms gives us $-1x \sqrt{775 x y}$. The expression now becomes $-1x \sqrt{775 x y} + 8 x \sqrt{31 x y}$. This initial combination significantly simplifies the expression, reducing the number of terms we need to work with. Combining like terms is a fundamental technique in algebra, and its application here streamlines the simplification process. By focusing on identifying and combining terms with identical radical components, we lay the groundwork for further simplification and make the overall task more manageable. This step-by-step approach ensures that each transformation is clear and easy to follow, building confidence in your ability to simplify radical expressions.

Now that we've combined the like terms, the next step involves simplifying the radicals individually. We have two radical terms in our expression: $-1x \sqrt{775 x y}$ and $8 x \sqrt{31 x y}$. Let's focus on the first term, $-1x \sqrt{775 x y}$. As we discussed earlier, we can simplify $\sqrt{775 x y}$ by factoring 775 into its prime factors. The prime factorization of 775 is $5^2 \cdot 31$. Therefore, we can rewrite $\sqrt{775 x y}$ as $\sqrt{5^2 \cdot 31 x y}$. The presence of the perfect square $5^2$ allows us to simplify the radical. We take the square root of $5^2$, which is 5, and move it outside the radical, resulting in $5 \sqrt{31 x y}$. Substituting this back into our term, we get $-1x \cdot 5 \sqrt{31 x y}$, which simplifies to $-5x \sqrt{31 x y}$. Next, we consider the second term, $8 x \sqrt{31 x y}$. Since 31 is a prime number and x and y are variables, the radical $\sqrt{31 x y}$ cannot be simplified further. Therefore, this term remains as $8 x \sqrt{31 x y}$. Simplifying radicals by factoring and identifying perfect squares is a crucial technique in handling radical expressions. This step not only makes the expression more concise but also reveals its underlying structure. By breaking down the radicand into its prime factors, we can effectively identify and extract perfect squares, leading to a simplified form of the radical. This technique is applicable to a wide range of radical expressions and is a key skill in algebraic manipulation.

After simplifying the radicals, we can now combine the simplified terms. Our expression is currently $-5x \sqrt{31 x y} + 8 x \sqrt{31 x y}$. Notice that both terms now have the same radical part, $\sqrt{31 x y}$. This means they are like terms and can be combined by adding their coefficients. The coefficients are -5x and 8x. Adding these coefficients gives us $-5x + 8x = 3x$. Therefore, the combined term is $3x \sqrt{31 x y}$. This final combination completes the simplification process. We have successfully reduced the original expression to a single term, making it much easier to understand and work with. Combining like terms after simplifying radicals is a crucial step in arriving at the most simplified form of the expression. This step highlights the importance of identifying and grouping terms with identical radical parts, which is a fundamental skill in algebraic simplification. By systematically simplifying radicals and then combining like terms, we can transform complex expressions into more manageable forms, facilitating further mathematical operations and analysis. The ability to efficiently simplify expressions is a cornerstone of algebraic proficiency and is essential for success in more advanced mathematical topics.

After performing all the necessary simplifications, the final simplified expression is $3x \sqrt{31 x y}$. This result represents the most concise and simplified form of the original expression, $7 x \sqrt{775 x y}+8 x \sqrt{31 x y}-8 x \sqrt{775 x y}$. Throughout this process, we have applied several key techniques, including combining like terms and simplifying radicals by factoring out perfect squares. These techniques are fundamental to simplifying radical expressions and are widely applicable in various mathematical contexts. The ability to reduce complex expressions to their simplest forms is crucial for problem-solving and further mathematical analysis. A simplified expression not only provides a clearer understanding of the relationship between variables but also facilitates subsequent operations, such as solving equations or graphing functions. By mastering the process of simplification, you gain a powerful tool that enhances your mathematical capabilities and confidence. This final simplified expression serves as a testament to the effectiveness of systematic simplification techniques and underscores the importance of these skills in mathematical proficiency.

In conclusion, mastering the art of simplifying radical expressions is a critical skill in mathematics. Through the step-by-step process outlined in this guide, we have demonstrated how to effectively combine radical terms using indicated operations, assuming all variables are positive. The initial expression, $7 x \sqrt{775 x y}+8 x \sqrt{31 x y}-8 x \sqrt{775 x y}$, was systematically simplified to its final form: $3x \sqrt{31 x y}$. This simplification involved several key steps, including identifying and combining like terms, factoring radicands to find perfect squares, and extracting those perfect squares from the radicals. Each step is essential in reducing the complexity of the expression and arriving at its most simplified form. The techniques discussed in this article are not only applicable to this specific problem but also serve as a foundation for simplifying a wide range of radical expressions. By consistently applying these methods, you can confidently tackle more complex mathematical problems and gain a deeper understanding of algebraic manipulation. The ability to simplify radical expressions is a valuable asset in various fields, including mathematics, physics, engineering, and computer science. As you continue your mathematical journey, remember that practice and consistent application of these techniques will solidify your skills and enhance your problem-solving abilities. This mastery will empower you to approach mathematical challenges with confidence and precision.

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