Simplifying Radical Expressions A Step-by-Step Guide To 2√72+7√98-2√50+3√32

Simplifying radical expressions is a fundamental skill in mathematics, particularly in algebra and precalculus. Mastery of this skill allows for more manageable calculations and a clearer understanding of numerical relationships. In this comprehensive guide, we will delve into the process of simplifying radical expressions, using the specific example of 2√72 + 7√98 - 2√50 + 3√32 to illustrate each step. Before diving into the example, it’s crucial to understand the basic principles and properties of radicals. A radical expression consists of a radical symbol (√), a radicand (the number under the radical), and an index (the degree of the root). For square roots, the index is implicitly 2. Simplifying a radical means reducing the radicand to its simplest form, where no perfect square factors remain. This involves factoring the radicand, identifying perfect square factors, and extracting them from under the radical sign. Perfect squares are numbers that can be expressed as the square of an integer, such as 4 (2^2), 9 (3^2), 16 (4^2), and so on. Simplifying radical expressions is not just a mathematical exercise; it is a practical tool that can be applied in various fields, such as physics, engineering, and computer science. Understanding how to manipulate radicals allows for more efficient problem-solving and a deeper understanding of mathematical concepts. Furthermore, simplifying radicals is often a necessary step in solving equations, graphing functions, and performing other algebraic operations. The process of simplifying radicals reinforces key mathematical concepts such as factoring, prime numbers, and the properties of exponents. By mastering these concepts, students can build a stronger foundation for more advanced mathematical topics. So, let’s embark on this detailed journey to unravel the complexities of radical simplification and gain a solid understanding of the process.

Understanding the Basics of Radicals

To effectively simplify the expression 2√72 + 7√98 - 2√50 + 3√32, it is essential to have a firm grasp of the fundamental properties of radicals. Understanding radicals begins with recognizing that a radical is essentially the inverse operation of exponentiation. The square root of a number x (√x) is a value that, when multiplied by itself, equals x. For example, the square root of 9 is 3 because 3 * 3 = 9. Similarly, the cube root of a number x (∛x) is a value that, when multiplied by itself three times, equals x. The nth root of a number x is written as ⁿ√x, where n is the index of the radical. The most common type of radical is the square root, which has an implied index of 2. The radicand is the number or expression under the radical symbol. Simplifying a radical expression involves breaking down the radicand into its prime factors and identifying any perfect square factors. A perfect square is a number that can be expressed as the square of an integer, such as 4 (2^2), 9 (3^2), 16 (4^2), 25 (5^2), and so on. The key to simplifying radicals is to factor out these perfect squares from the radicand. The properties of radicals are essential tools in this process. One crucial property is the product property of radicals, which states that the square root of a product is equal to the product of the square roots: √(ab) = √a * √b. This property allows us to separate the radicand into factors and simplify each factor individually. Another important property is the quotient property of radicals, which states that the square root of a quotient is equal to the quotient of the square roots: √(a/b) = √a / √b. This property is particularly useful when dealing with fractions under the radical. To simplify a radical expression, we follow a systematic approach: First, we factor the radicand into its prime factors. Next, we identify any perfect square factors. Then, we rewrite the radicand as a product of the perfect square factors and the remaining factors. Finally, we apply the product property of radicals to separate the perfect square factors and simplify them. Let's consider a simple example: √20. We can factor 20 as 4 * 5, where 4 is a perfect square (2^2). Using the product property, we can rewrite √20 as √(4 * 5) = √4 * √5 = 2√5. Thus, the simplified form of √20 is 2√5. Understanding these basics is crucial for tackling more complex radical expressions, such as the one we will simplify in the following sections.

Step-by-Step Simplification of 2√72

The first term we need to simplify in the expression 2√72 + 7√98 - 2√50 + 3√32 is 2√72. Simplifying 2√72 involves breaking down the radicand, 72, into its prime factors and identifying any perfect square factors. Prime factorization is the process of expressing a number as a product of its prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. The prime factors of 72 are 2 and 3. To find the prime factorization of 72, we can use a factor tree: 72 = 2 * 36 = 2 * 2 * 18 = 2 * 2 * 2 * 9 = 2 * 2 * 2 * 3 * 3. So, the prime factorization of 72 is 2^3 * 3^2. Now, we rewrite the radical expression using the prime factorization: 2√72 = 2√(2^3 * 3^2). The next step is to identify any perfect square factors. We know that 2^2 = 4 and 3^2 = 9 are perfect squares. We can rewrite 2^3 as 2^2 * 2. So, the expression becomes: 2√(2^2 * 2 * 3^2). Using the product property of radicals, we can separate the perfect square factors: 2√(2^2 * 2 * 3^2) = 2 * √2^2 * √2 * √3^2. Now, we simplify the square roots of the perfect squares: 2 * √2^2 * √2 * √3^2 = 2 * 2 * √2 * 3. Multiplying the constants, we get: 2 * 2 * √2 * 3 = 12√2. Therefore, the simplified form of 2√72 is 12√2. This step-by-step approach ensures that we correctly identify and extract all perfect square factors from the radicand. Simplifying radicals in this manner is crucial for performing further operations and for expressing the final result in its simplest form. Understanding this process lays the foundation for simplifying the other terms in the expression and ultimately solving the entire problem. The ability to break down numbers into their prime factors and identify perfect squares is a fundamental skill in simplifying radical expressions and is essential for success in algebra and beyond. By following this method, we can confidently simplify any radical expression, no matter how complex it may seem at first glance.

Step-by-Step Simplification of 7√98

Next, we move on to simplifying 7√98, another term in the original expression 2√72 + 7√98 - 2√50 + 3√32. Similar to the previous step, we begin by finding the prime factorization of the radicand, which is 98. Prime factorization involves breaking down the number into its prime factors. We can find the prime factors of 98 as follows: 98 = 2 * 49 = 2 * 7 * 7. Therefore, the prime factorization of 98 is 2 * 7^2. Now, we rewrite the radical expression using the prime factorization: 7√98 = 7√(2 * 7^2). We identify the perfect square factor in the radicand, which is 7^2 = 49. We can rewrite the expression as: 7√(2 * 7^2). Using the product property of radicals, we separate the perfect square factor: 7√(2 * 7^2) = 7 * √2 * √7^2. Simplifying the square root of the perfect square, we get: 7 * √2 * √7^2 = 7 * √2 * 7. Multiplying the constants, we have: 7 * √2 * 7 = 49√2. Thus, the simplified form of 7√98 is 49√2. This process of identifying and extracting perfect square factors is crucial for simplifying radical expressions. By breaking down the radicand into its prime factors, we can easily spot the perfect squares and simplify the expression. The product property of radicals allows us to separate the factors and simplify them individually, making the process more manageable. Simplifying 7√98 to 49√2 is a significant step in simplifying the overall expression. It allows us to combine like terms later on, which is essential for obtaining the final simplified form. Mastering the technique of simplifying radicals through prime factorization and the application of radical properties is a fundamental skill in algebra. This skill is not only useful for simplifying expressions but also for solving equations and tackling more advanced mathematical problems. By practicing these steps, you can develop a strong foundation in radical simplification and enhance your overall mathematical proficiency. The ability to efficiently simplify radicals is a valuable asset in various mathematical contexts.

Step-by-Step Simplification of 2√50

Now, let’s focus on simplifying 2√50, the third term in the expression 2√72 + 7√98 - 2√50 + 3√32. As we have done with the previous terms, the first step in simplifying this radical expression is to find the prime factorization of the radicand, which is 50. Prime factorization involves expressing the number as a product of its prime factors. We can break down 50 as follows: 50 = 2 * 25 = 2 * 5 * 5. Therefore, the prime factorization of 50 is 2 * 5^2. Rewriting the radical expression using the prime factorization, we have: 2√50 = 2√(2 * 5^2). We identify the perfect square factor in the radicand, which is 5^2 = 25. We can rewrite the expression as: 2√(2 * 5^2). Applying the product property of radicals, we separate the perfect square factor: 2√(2 * 5^2) = 2 * √2 * √5^2. Simplifying the square root of the perfect square, we get: 2 * √2 * √5^2 = 2 * √2 * 5. Multiplying the constants, we have: 2 * √2 * 5 = 10√2. Thus, the simplified form of 2√50 is 10√2. This step highlights the consistent approach we use to simplify radical expressions. By breaking down the radicand into its prime factors, identifying perfect square factors, and applying the product property of radicals, we can systematically simplify any radical term. Simplifying 2√50 to 10√2 is an important step in simplifying the overall expression. It allows us to combine this term with other terms that have the same radical component, which is essential for obtaining the final simplified form. The process of simplifying radicals is a fundamental skill in algebra and is used extensively in various mathematical contexts. By mastering this skill, you can confidently tackle more complex problems and gain a deeper understanding of mathematical concepts. The ability to efficiently simplify radicals is a valuable asset in problem-solving and mathematical reasoning. This step-by-step approach not only simplifies the expression but also reinforces the underlying principles of radical simplification, making it easier to apply these techniques to other problems.

Step-by-Step Simplification of 3√32

Finally, we need to simplify 3√32, the last term in the expression 2√72 + 7√98 - 2√50 + 3√32. As with the previous terms, we begin by finding the prime factorization of the radicand, which is 32. Prime factorization involves breaking down the number into its prime factors. We can factor 32 as follows: 32 = 2 * 16 = 2 * 2 * 8 = 2 * 2 * 2 * 4 = 2 * 2 * 2 * 2 * 2. Therefore, the prime factorization of 32 is 2^5. Now, we rewrite the radical expression using the prime factorization: 3√32 = 3√(2^5). We need to identify perfect square factors within 2^5. We can rewrite 2^5 as 2^4 * 2, where 2^4 is a perfect square (since 2^4 = 16). So, the expression becomes: 3√(2^4 * 2). Using the product property of radicals, we can separate the perfect square factor: 3√(2^4 * 2) = 3 * √2^4 * √2. Now, we simplify the square root of the perfect square: 3 * √2^4 * √2 = 3 * 2^2 * √2. Simplifying further, we get: 3 * 4 * √2 = 12√2. Therefore, the simplified form of 3√32 is 12√2. This step-by-step process demonstrates how to handle radicands with higher powers. By breaking down the radicand into its prime factors and identifying the highest possible perfect square factor, we can efficiently simplify the radical expression. Simplifying 3√32 to 12√2 is the final step in simplifying each individual term in the original expression. This allows us to combine like terms in the next step, which is crucial for obtaining the final simplified form. The consistency of this approach across all terms reinforces the importance of prime factorization and the application of radical properties in simplifying radical expressions. The ability to handle higher powers within radicands is a valuable skill in algebra, and mastering this technique will enhance your overall mathematical proficiency. By following these steps, you can confidently simplify any radical expression, regardless of the complexity of the radicand.

Combining Like Terms

After simplifying each term individually, we now have the expression in the form 12√2 + 49√2 - 10√2 + 12√2. The next crucial step is to combine like terms. In the context of radical expressions, like terms are those that have the same radical component. In this case, all the terms have the same radical component, which is √2. To combine like terms, we simply add or subtract the coefficients (the numbers in front of the radical) while keeping the radical component the same. This is similar to combining like terms in algebraic expressions, where you would combine terms with the same variable. So, we have: 12√2 + 49√2 - 10√2 + 12√2. We combine the coefficients: (12 + 49 - 10 + 12)√2. Performing the addition and subtraction: (61 - 10 + 12)√2 = (51 + 12)√2 = 63√2. Therefore, the simplified form of the expression 12√2 + 49√2 - 10√2 + 12√2 is 63√2. Combining like terms is a fundamental step in simplifying any algebraic expression, including those involving radicals. It allows us to reduce the expression to its simplest form, making it easier to work with in further calculations or applications. This step highlights the importance of simplifying each term before attempting to combine them. If we had not simplified the radicals first, we would not have been able to identify the like terms and combine them effectively. The process of combining like terms reinforces the concept of treating radicals as single entities, similar to variables in algebraic expressions. By adding or subtracting the coefficients, we are essentially performing arithmetic operations on the multiples of the radical component. Mastering the skill of combining like terms is essential for success in algebra and beyond. It is a fundamental technique that is used in various mathematical contexts, including solving equations, simplifying expressions, and working with functions. By practicing this step, you can develop a strong foundation in algebraic manipulation and enhance your overall mathematical proficiency.

Final Simplified Expression

After performing all the necessary steps, we have arrived at the final simplified expression. Starting with the original expression 2√72 + 7√98 - 2√50 + 3√32, we systematically simplified each term: * 2√72 simplified to 12√2 * 7√98 simplified to 49√2 * 2√50 simplified to 10√2 * 3√32 simplified to 12√2. We then combined the like terms: 12√2 + 49√2 - 10√2 + 12√2. By adding and subtracting the coefficients, we obtained: (12 + 49 - 10 + 12)√2 = 63√2. Therefore, the final simplified expression is 63√2. This result represents the simplest form of the original expression, where all radicals have been simplified, and like terms have been combined. The final simplified expression, 63√2, is a single term, making it easier to work with in further calculations or applications. This process demonstrates the importance of following a systematic approach when simplifying radical expressions. By breaking down the problem into smaller steps, we can effectively manage the complexity and arrive at the correct answer. The simplified expression not only looks cleaner but also provides a clearer understanding of the numerical value. It is often easier to compare and manipulate simplified expressions than their more complex counterparts. Simplifying radical expressions is a crucial skill in algebra and is used extensively in various mathematical contexts. From solving equations to working with functions, the ability to simplify radicals is a valuable asset. By mastering this skill, you can enhance your mathematical proficiency and gain a deeper understanding of algebraic concepts. The final result, 63√2, is a concise and elegant representation of the original expression, highlighting the power of simplification in mathematics. This process not only provides the answer but also reinforces the fundamental principles of radical simplification, making it easier to apply these techniques to other problems. In summary, the journey from the initial complex expression to the final simplified form showcases the importance of a step-by-step approach, the application of radical properties, and the combination of like terms in simplifying radical expressions.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill in mathematics that requires a systematic approach and a solid understanding of fundamental principles. Throughout this guide, we have demonstrated a step-by-step method for simplifying radical expressions using the example of 2√72 + 7√98 - 2√50 + 3√32. We began by understanding the basics of radicals, including the definition of a radicand, index, and perfect squares. Then, we broke down each term in the expression, finding the prime factorization of the radicands and identifying perfect square factors. We applied the product property of radicals to separate the perfect square factors and simplify them individually. After simplifying each term, we combined like terms by adding or subtracting their coefficients. Finally, we arrived at the simplified expression, 63√2. This process highlights the importance of several key concepts: * Prime factorization: Breaking down a number into its prime factors is essential for identifying perfect square factors. * Product property of radicals: This property allows us to separate the radicand into factors and simplify each factor individually. * Combining like terms: This step is crucial for reducing the expression to its simplest form. Mastering the simplification of radical expressions not only enhances your algebraic skills but also provides a solid foundation for more advanced mathematical topics. The ability to simplify radicals is essential in various mathematical contexts, including solving equations, working with functions, and tackling complex problems in calculus and beyond. By following the steps outlined in this guide, you can confidently simplify any radical expression, no matter how complex it may seem. The simplified form not only looks cleaner but also provides a clearer understanding of the numerical value. The skill of simplifying radical expressions is a valuable asset in problem-solving and mathematical reasoning. It allows you to manipulate expressions more efficiently and arrive at solutions more effectively. The journey from the initial complex expression to the final simplified form showcases the power of a systematic approach and the importance of understanding fundamental mathematical principles. By practicing these techniques, you can develop a strong foundation in algebra and enhance your overall mathematical proficiency. So, continue to practice and apply these skills to various problems, and you will become proficient in simplifying radical expressions.