Expanding And Simplifying (x√7 - 3√8)(x√7 - 3√8) A Step By Step Guide
Introduction: Understanding the Product of Binomials with Radicals
In mathematics, understanding how to multiply binomials, especially those involving radicals, is a fundamental skill. This article delves into the step-by-step process of expanding the product (x√7 - 3√8)(x√7 - 3√8). We will explore the techniques required to simplify expressions with square roots and combine like terms effectively. Mastering these techniques is not only crucial for algebraic manipulations but also essential for various applications in geometry, calculus, and other advanced mathematical fields. Let's break down each step of this process to ensure a clear and comprehensive understanding.
At the heart of this problem lies the concept of binomial multiplication. When we multiply two binomials together, we are essentially distributing each term of the first binomial across the terms of the second binomial. The most common method for achieving this is the FOIL method, which stands for First, Outer, Inner, and Last. This mnemonic helps us remember to multiply every term correctly. For example, in the expression (a + b)(c + d), we would multiply the First terms (a and c), then the Outer terms (a and d), followed by the Inner terms (b and c), and finally the Last terms (b and d). This gives us the expanded form ac + ad + bc + bd. When dealing with radicals, we must also remember the properties of square roots. The square root of a product is the product of the square roots, i.e., √(ab) = √a * √b. Additionally, √a * √a = a. These rules are crucial for simplifying expressions and combining like terms effectively. The combination of these algebraic and radical properties allows us to tackle complex expressions with confidence. By applying these principles methodically, we can accurately expand and simplify the given product, gaining valuable insights into algebraic manipulation and problem-solving in mathematics.
Step-by-Step Expansion of (x√7 - 3√8)(x√7 - 3√8)
To find the product of (x√7 - 3√8)(x√7 - 3√8), we will systematically apply the principles of binomial multiplication and radical simplification. This process involves several key steps, each building upon the previous one to ensure accuracy and clarity. Let's delve into each step in detail.
First, we apply the distributive property, commonly remembered by the acronym FOIL (First, Outer, Inner, Last), to expand the product. This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Breaking this down:
- First: Multiply the first terms of each binomial:
(x√7) * (x√7) - Outer: Multiply the outer terms:
(x√7) * (-3√8) - Inner: Multiply the inner terms:
(-3√8) * (x√7) - Last: Multiply the last terms:
(-3√8) * (-3√8)
This process gives us the expanded form: (x√7)(x√7) - (x√7)(3√8) - (3√8)(x√7) + (3√8)(3√8). Each term now needs to be simplified further.
Next, we simplify each term by performing the multiplications. The first term, (x√7)(x√7), simplifies to x² * (√7)², which is 7x². The outer term, -(x√7)(3√8), simplifies to -3x√(7*8), which is -3x√56. Similarly, the inner term, -(3√8)(x√7), simplifies to -3x√56. The last term, (3√8)(3√8), simplifies to 9 * (√8)², which is 9 * 8 = 72. Combining these simplified terms gives us: 7x² - 3x√56 - 3x√56 + 72. This intermediate step is crucial for visualizing the components before further simplification.
Now, we combine like terms. In this case, the terms -3x√56 and -3x√56 are like terms because they both contain the variable x and the radical √56. Combining them yields -6x√56. This simplification reduces the number of terms and makes the expression more manageable. The expression now looks like this: 7x² - 6x√56 + 72. This combination is a key step in simplifying the overall expression and preparing it for further simplification.
Finally, we simplify the radical term √56. We can break down 56 into its prime factors: 56 = 2 * 2 * 2 * 7. This allows us to rewrite √56 as √(2² * 2 * 7). Applying the property of square roots, we can take out the perfect square 2², resulting in 2√(2 * 7), which simplifies to 2√14. Substituting this back into our expression, we get 7x² - 6x(2√14) + 72. Further simplifying, we have 7x² - 12x√14 + 72. This final simplification provides the most concise and understandable form of the expression. By following these steps meticulously, we have successfully expanded and simplified the original product.
Detailed Simplification of the Radical Term √56
Simplifying radical terms is a critical aspect of algebraic manipulation, especially when dealing with expressions that involve square roots. The term √56 is a prime example of a radical that can be further simplified to reveal its underlying structure and make calculations easier. To simplify √56, we need to break it down into its prime factors and identify any perfect square factors. This process allows us to express the radical in its simplest form, which is essential for combining like terms and performing other algebraic operations.
First, we factorize 56 into its prime factors. Prime factorization involves expressing a number as a product of its prime numbers. The prime factors of 56 are 2 and 7. Breaking 56 down, we find that 56 = 2 * 28 = 2 * 2 * 14 = 2 * 2 * 2 * 7. So, the prime factorization of 56 is 2³ * 7. This step is crucial because it allows us to see all the individual components of the number under the square root and identify any perfect squares.
Next, we rewrite the square root using the prime factors. Substituting the prime factorization of 56 into the square root, we have √56 = √(2³ * 7). This form is more conducive to simplification because it highlights the powers of the prime factors. Now we can apply the properties of square roots to separate and simplify the factors.
Then, we identify and extract perfect square factors. A perfect square is a number that can be expressed as the square of an integer (e.g., 4, 9, 16). In the prime factorization 2³ * 7, we can rewrite 2³ as 2² * 2. Thus, we have √(2³ * 7) = √(2² * 2 * 7). The perfect square here is 2². We can take the square root of 2², which is 2. This allows us to extract the perfect square from under the radical, simplifying the expression.
Finally, we simplify the expression by taking the square root of the perfect square and leaving the remaining factors under the radical. We rewrite √(2² * 2 * 7) as √(2²) * √(2 * 7). The square root of 2² is 2, and the square root of 2 * 7 is √14. Therefore, the simplified form of √56 is 2√14. This simplified radical term is much easier to work with in algebraic expressions. By following these steps, we have successfully simplified the radical √56, making it clearer and more manageable for further calculations.
Combining Like Terms: -3x√56 - 3x√56
In algebraic expressions, combining like terms is a fundamental step in simplification. Like terms are those that have the same variables raised to the same powers. When dealing with expressions involving radicals, this principle extends to terms with the same radical component. In our expression, we encountered the terms -3x√56 and -3x√56. These are like terms because they both contain the variable x and the radical √56. The process of combining them involves adding their coefficients while keeping the variable and radical component the same. Understanding this process is crucial for simplifying complex expressions and solving equations efficiently.
To identify like terms, we look for terms that have the same variable part and the same radical part. In the expression -3x√56 - 3x√56, both terms have x as the variable part and √56 as the radical part. This means they are indeed like terms and can be combined. Recognizing like terms is the first step in simplifying any algebraic expression. It helps in reducing the complexity and makes further calculations easier.
Next, we add the coefficients of the like terms. The coefficients are the numerical parts of the terms. In this case, the coefficients are -3 and -3. Adding these coefficients together gives us -3 + (-3) = -6. This arithmetic operation is straightforward but essential. It forms the basis of combining like terms and reducing the expression to its simplest form. This is a fundamental arithmetic operation that is used extensively in algebra.
Then, we combine the terms by keeping the variable and radical part the same while using the new coefficient. After adding the coefficients, we have -6. We then combine this coefficient with the variable x and the radical √56. This results in the simplified term -6x√56. This step is the culmination of the process of combining like terms. It simplifies the expression by reducing the number of terms while maintaining the integrity of the original mathematical relationship.
Therefore, the combination of -3x√56 and -3x√56 results in -6x√56. This simplified term is easier to work with in subsequent steps of simplification or problem-solving. Combining like terms is a powerful technique that streamlines algebraic expressions and makes them more manageable. It reduces the expression to its most concise form, which is crucial for solving equations and further mathematical analysis. This process is a cornerstone of algebraic manipulation and is used extensively in more advanced mathematical concepts.
Final Simplified Form: 7x² - 12x√14 + 72
After expanding, simplifying the radical term, and combining like terms, we arrive at the final simplified form of the expression (x√7 - 3√8)(x√7 - 3√8). This simplified form is crucial because it presents the expression in its most understandable and manageable state. The final form not only makes it easier to perform further calculations but also provides a clear view of the relationship between the variables and constants involved. Let's walk through the final steps to ensure we understand how we reached this conclusion.
We started with the expression 7x² - 6x√56 + 72. We had previously simplified the radical √56 to 2√14. Now, we substitute the simplified radical back into the expression. Replacing √56 with 2√14 in the expression, we get 7x² - 6x(2√14) + 72. This substitution is a critical step because it replaces the complex radical with its simplified version, making the expression easier to manage.
Next, we perform the multiplication in the term -6x(2√14). Multiplying the coefficient -6 by 2 gives us -12. So, the term becomes -12x√14. The expression now looks like 7x² - 12x√14 + 72. This multiplication step further simplifies the expression by removing the parentheses and combining numerical factors.
The resulting expression, 7x² - 12x√14 + 72, is the final simplified form. There are no more like terms to combine, and the radical is in its simplest form. This expression is a quadratic expression involving a radical term. It represents the most concise and understandable version of the original product. This final form is crucial for various mathematical operations, such as solving equations, graphing functions, and further algebraic analysis.
In conclusion, the final simplified form of the product (x√7 - 3√8)(x√7 - 3√8) is 7x² - 12x√14 + 72. This process demonstrates the importance of methodical algebraic manipulation, including expanding products, simplifying radicals, and combining like terms. By following these steps, we can transform complex expressions into more manageable forms, making them easier to understand and use in various mathematical contexts.
Conclusion: Mastering Algebraic Manipulation
In summary, the product (x√7 - 3√8)(x√7 - 3√8) simplifies to 7x² - 12x√14 + 72. This simplification process illustrates several key concepts in algebraic manipulation, including the distributive property, simplification of radicals, and combining like terms. Mastering these techniques is essential for success in algebra and beyond. By understanding and applying these principles, one can confidently tackle complex algebraic expressions and simplify them into manageable forms. The step-by-step approach detailed in this article provides a clear and concise method for handling such problems, ensuring accuracy and comprehension.
The ability to expand and simplify algebraic expressions is a fundamental skill in mathematics. It is the foundation upon which more advanced mathematical concepts are built. The process involves breaking down complex expressions into smaller, more manageable parts, and then applying specific rules and techniques to simplify each part. This includes understanding the distributive property, which allows us to multiply binomials, and the properties of radicals, which enable us to simplify square roots. Furthermore, the ability to recognize and combine like terms is crucial for reducing expressions to their simplest form. This comprehensive approach not only simplifies the expression but also enhances one's problem-solving skills and mathematical intuition.
Understanding and applying these algebraic techniques are crucial for a variety of mathematical contexts. These skills are not only useful in algebra but also in calculus, geometry, and other advanced mathematical fields. For instance, in calculus, simplifying expressions is often a necessary step before differentiation or integration. In geometry, these skills can be used to solve problems involving areas and volumes. Moreover, in real-world applications, algebraic manipulation is essential for modeling and solving problems in physics, engineering, and economics. Therefore, mastering these techniques is not just an academic exercise but a valuable tool for problem-solving in various domains. The ability to manipulate algebraic expressions effectively opens doors to a deeper understanding of mathematical concepts and their practical applications.
In conclusion, the simplification of (x√7 - 3√8)(x√7 - 3√8) to 7x² - 12x√14 + 72 serves as a powerful illustration of the importance of algebraic manipulation. This process involves a series of steps, each building upon the previous one to achieve the final simplified form. From expanding the product using the distributive property to simplifying radicals and combining like terms, each step is crucial for achieving the correct result. Mastering these techniques not only enhances one's algebraic skills but also provides a solid foundation for tackling more complex mathematical problems. The ability to simplify expressions efficiently and accurately is a valuable asset in mathematics and various fields that rely on mathematical principles.