Simplifying Radical Expressions A Step-by-Step Guide

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At the heart of this mathematical exploration lies the challenge of simplifying a product involving radicals and variables. Specifically, we are tasked with simplifying the expression: (√10x⁴ - x√5x²)(2√15x⁴ + √3x³), given the condition that x ≥ 0. This condition is crucial because it ensures that the square roots are defined within the real number system. The expression combines square roots, variables raised to different powers, and arithmetic operations, making it a comprehensive test of algebraic manipulation skills. Our goal is to navigate through this complex expression, applying the rules of radicals and exponents, to arrive at a simplified form. This process will involve several key steps, including distributing the terms, simplifying radicals, and combining like terms. The final simplified expression will not only be more concise but also reveal the underlying structure of the original expression. Understanding this simplification process is fundamental in algebra, as it lays the groundwork for solving more complex equations and understanding advanced mathematical concepts. The problem requires a strong understanding of radical properties and the ability to apply these properties in conjunction with algebraic manipulation techniques. By successfully simplifying this expression, we not only arrive at the correct answer but also reinforce our grasp of these essential mathematical skills. The journey of simplification is as important as the final result, as it provides valuable practice in problem-solving and mathematical reasoning. This particular problem stands out due to its combination of radicals, variables, and the need for careful application of distribution and simplification rules. It serves as an excellent example of how algebraic expressions can be manipulated to reveal their underlying simplicity. In the following sections, we will delve into the step-by-step solution, highlighting the key concepts and techniques used in the simplification process.

Step-by-Step Solution: Unraveling the Complexity

To tackle the expression (√10x⁴ - x√5x²)(2√15x⁴ + √3x³), a systematic approach is essential. We will break down the simplification process into manageable steps, ensuring clarity and accuracy along the way. Our primary strategy involves applying the distributive property (often referred to as the FOIL method) to expand the product, followed by simplifying each term individually using the properties of radicals and exponents. Let's embark on this mathematical journey, step by meticulous step.

1. Expanding the Product: Applying the Distributive Property

The first step in simplifying the expression is to expand the product using the distributive property. This involves multiplying each term in the first parenthesis by each term in the second parenthesis. The distributive property, in this context, is the key to unlocking the expression's potential for simplification. By carefully applying this property, we transform the product into a sum of individual terms, each of which can then be simplified independently. The expansion process is as follows:

(√10x⁴ - x√5x²)(2√15x⁴ + √3x³) expands to:

(√10x⁴)(2√15x⁴) + (√10x⁴)(√3x³) - (x√5x²)(2√15x⁴) - (x√5x²)(√3x³)

This expansion results in four terms, each a product of radicals and variables. The next step is to simplify each of these terms individually. This involves combining the radicals and applying the properties of exponents to the variables. The distributive property is a cornerstone of algebra, and its correct application here is crucial for the subsequent steps. By expanding the product, we have effectively transformed a complex expression into a series of simpler terms, each of which can be addressed using familiar techniques. The expansion process sets the stage for the simplification of radicals and the combination of like terms, ultimately leading to the simplified form of the expression. The meticulous application of the distributive property ensures that no terms are missed and that the expansion is accurate, laying a solid foundation for the rest of the solution.

2. Simplifying Each Term: Taming the Radicals and Exponents

Now that we have expanded the product, the next crucial step is to simplify each term individually. This involves a combination of simplifying radicals and applying the properties of exponents. Each term presents a unique challenge, requiring a careful and methodical approach. The goal is to express each term in its simplest form, which will make the subsequent combination of like terms much easier.

Let's break down the simplification of each term:

  • (√10x⁴)(2√15x⁴): Combine the radicals and variables: 2√(10 * 15) * x^(4+4) = 2√150 * x^8. Simplify the radical: 2√(25 * 6) * x^8 = 2 * 5√6 * x^8 = 10x^8√6
  • (√10x⁴)(√3x³): Combine the radicals and variables: √(10 * 3) * x^(4+3) = √30 * x^7 = x^7√30
  • (x√5x²)(2√15x⁴): Combine the terms: 2x√(5 * 15) * x^(2+4) = 2x√75 * x^6. Simplify the radical: 2x√(25 * 3) * x^6 = 2x * 5√3 * x^6 = 10x^7√3
  • (x√5x²)(√3x³): Combine the terms: x√(5 * 3) * x^(2+3) = x√15 * x^5 = x^6√15

Each of these simplifications involves breaking down the radicals into their prime factors and extracting any perfect squares. The properties of exponents are also used to combine the variable terms. This step-by-step approach ensures that no simplification is overlooked and that each term is expressed in its most basic form. By simplifying each term individually, we have paved the way for the final step: combining like terms. The simplified terms now present a clearer picture of the expression's structure, making the final combination a more straightforward process. The meticulous simplification of each term is a testament to the importance of attention to detail in mathematical problem-solving.

3. Combining Like Terms: The Final Synthesis

After simplifying each term, the final step in this mathematical journey is to combine like terms. Like terms are those that have the same variable and radical components. This process involves identifying terms that share these characteristics and then adding or subtracting their coefficients. Combining like terms is a fundamental algebraic technique that allows us to express the expression in its most concise and understandable form. The simplified expression not only looks cleaner but also reveals the underlying mathematical relationships more clearly.

From the previous step, we have the following simplified terms:

  • 10x^8√6
  • x^7√30
  • -10x^7√3
  • -x^6√15

Now, we look for terms that have the same radical and variable components. In this case, there are no like terms that can be combined. Each term has a unique combination of radicals and variables, which means that the expression is already in its simplest form. This outcome highlights the importance of careful simplification in the previous steps. If any errors had been made in simplifying the radicals or exponents, it might have led to incorrect identification of like terms or missed opportunities for combination. The absence of like terms to combine is not a failure but rather a confirmation that the expression has been thoroughly simplified.

Therefore, the final simplified expression is:

10x^8√6 + x^7√30 - 10x^7√3 - x^6√15

This expression represents the most simplified form of the original product, given the condition that x ≥ 0. The combination of like terms, or in this case, the lack thereof, marks the culmination of the simplification process. The final expression is a testament to the power of algebraic manipulation and the importance of following a systematic approach. The journey from the initial complex product to this simplified expression has involved the application of several key mathematical concepts, including the distributive property, the simplification of radicals, and the combination of like terms. The final result is not only a mathematical answer but also a reflection of the problem-solving skills and mathematical reasoning that have been employed.

Final Answer: Presenting the Simplified Form

After a meticulous step-by-step simplification process, we arrive at the final answer. The simplified form of the expression (√10x⁴ - x√5x²)(2√15x⁴ + √3x³), given the condition x ≥ 0, is:

10x^8√6 + x^7√30 - 10x^7√3 - x^6√15

This expression represents the most concise and understandable form of the original product. The journey to this final answer has involved several key steps, each requiring careful attention to detail and a strong understanding of algebraic principles. The process began with expanding the product using the distributive property, followed by simplifying each term individually by applying the properties of radicals and exponents. Finally, we attempted to combine like terms, but in this case, there were no like terms to combine, indicating that the expression was already in its simplest form.

This final answer not only provides the solution to the problem but also highlights the power of algebraic manipulation in simplifying complex expressions. The simplified form reveals the underlying structure of the original product, making it easier to analyze and understand. The journey of simplification is a valuable exercise in mathematical reasoning and problem-solving, reinforcing key concepts and techniques that are essential in algebra and beyond. The final answer is a testament to the importance of a systematic approach, careful execution, and a solid understanding of mathematical principles. It represents not just a solution but also a demonstration of mathematical proficiency and the ability to transform complex expressions into simpler, more manageable forms. The final simplified expression stands as a clear and concise representation of the original product, a culmination of the simplification process and a testament to the beauty and power of mathematics.

Simplify the product: (√10x⁴ - x√5x²)(2√15x⁴ + √3x³), assuming x ≥ 0.

Simplifying Radical Expressions A Step-by-Step Guide