Simplifying √400x¹⁰⁰ A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to represent complex equations in a more manageable and understandable form. Simplifying radical expressions, in particular, often involves extracting perfect squares from under the radical sign. Today, we'll delve into the process of simplifying the expression √400x¹⁰⁰, breaking down each step to reveal the elegant solution. This exploration will not only provide the answer but also enhance your understanding of radical simplification techniques, a crucial aspect of algebra and beyond. Whether you're a student tackling algebra problems or simply a math enthusiast eager to expand your knowledge, this guide will provide a clear and concise pathway to mastering this essential skill. By the end of this article, you'll be well-equipped to tackle similar problems with confidence and precision, solidifying your foundation in mathematical simplification.

Before we dive into the specific problem, let's solidify our understanding of radical simplification. At its core, simplifying a radical expression means removing any perfect square factors from under the radical sign. A perfect square is a number or expression that can be obtained by squaring another number or expression. For example, 9 is a perfect square because it's the result of 3 squared (3² = 9). Similarly, x⁴ is a perfect square because it's the result of x² squared ((x²)² = x⁴). The key principle we'll be using is the property √(ab) = √a * √b, which allows us to separate the radical of a product into the product of radicals. This property is instrumental in isolating and extracting perfect square factors. Furthermore, we need to remember the fundamental relationship between square roots and exponents: √(x²) = |x|, where the absolute value ensures the result is non-negative. However, in many simplification problems, especially those encountered in introductory algebra, we often assume that variables represent non-negative numbers, which allows us to simplify √(x²) to simply x. This assumption streamlines the simplification process and helps us focus on the core concepts. Understanding these fundamental principles and properties is crucial for successfully simplifying radical expressions and laying a strong foundation for more advanced mathematical concepts.

Now, let's tackle the expression √400x¹⁰⁰ step by step. Our goal is to identify and extract any perfect square factors from under the radical. We'll start by examining the numerical coefficient, 400. Recognizing that 400 is a perfect square (20² = 400), we can immediately see a perfect square factor. Next, we turn our attention to the variable term, x¹⁰⁰. Here, we need to recall the rules of exponents. A variable raised to an even power is a perfect square because it can be expressed as the square of another variable term. Specifically, x¹⁰⁰ can be written as (x⁵⁰)², meaning it is indeed a perfect square. Now that we've identified the perfect square factors, we can rewrite the expression using the property √(ab) = √a * √b. This allows us to separate the radical into the product of two radicals: √400x¹⁰⁰ = √400 * √x¹⁰⁰. This separation is a crucial step in the simplification process, as it allows us to deal with each factor individually and apply the square root operation more easily. By carefully breaking down the expression in this way, we're setting the stage for the final simplification, which involves taking the square root of each perfect square factor. This methodical approach ensures accuracy and clarity in the solution.

The first part of our separated expression is √400. To simplify this, we need to find the square root of 400. As mentioned earlier, 400 is a perfect square, and its square root is 20. This is because 20 multiplied by itself (20 * 20) equals 400. Therefore, √400 simplifies directly to 20. This simplification highlights the importance of recognizing common perfect squares, such as 4, 9, 16, 25, 36, and so on. Being familiar with these squares allows for quick and efficient simplification of radical expressions. The square root of a perfect square is simply the number that, when multiplied by itself, gives the original perfect square. In this case, 20 is the number that, when multiplied by itself, gives 400. This straightforward calculation simplifies the numerical part of our expression, and we can now move on to simplifying the variable part. This step-by-step approach ensures that we address each component of the expression systematically, leading to a clear and accurate final result. Simplifying the numerical coefficient is a crucial step in the overall simplification process, and recognizing perfect squares is a valuable skill in this context.

Now, let's focus on the variable term, √x¹⁰⁰. To simplify this, we need to recall the relationship between square roots and exponents. The square root of a variable raised to an even power can be found by dividing the exponent by 2. In other words, √(xⁿ) = x^(n/2) when n is an even number. Applying this rule to our expression, we have √x¹⁰⁰. The exponent of x is 100, which is an even number. Dividing 100 by 2, we get 50. Therefore, √x¹⁰⁰ simplifies to x⁵⁰. This simplification demonstrates a key principle in dealing with radicals involving variables: even exponents under a square root can be easily handled by halving the exponent. It's crucial to remember this rule as it's a fundamental tool in simplifying algebraic expressions. The result, x⁵⁰, represents the simplified form of the variable term. This indicates that x⁵⁰ multiplied by itself (x⁵⁰ * x⁵⁰) equals x¹⁰⁰, confirming our simplification. Now that we've simplified both the numerical coefficient and the variable term, we're ready to combine the results to obtain the fully simplified expression. Understanding how to manipulate exponents within radicals is essential for mastering algebraic simplification, and this step provides a clear example of this process.

Having simplified both the numerical coefficient and the variable term, we now have 20 from √400 and x⁵⁰ from √x¹⁰⁰. To complete the simplification of the original expression, we simply multiply these two terms together. This is because we separated the original radical into the product of two radicals: √400x¹⁰⁰ = √400 * √x¹⁰⁰. Now, substituting the simplified forms, we get 20 * x⁵⁰, which is commonly written as 20x⁵⁰. This final step combines the individual simplifications into a single, concise expression. The process of combining simplified terms is a crucial part of radical simplification, as it brings together the results of the individual steps to form the complete answer. The simplified expression, 20x⁵⁰, is the most compact and understandable representation of the original expression, √400x¹⁰⁰. It clearly shows the relationship between the numerical coefficient and the variable term, and it eliminates the radical sign, which is the goal of simplification. This final step underscores the importance of a systematic approach to simplification, where each part of the expression is addressed individually and then combined to achieve the overall result. The ability to combine terms effectively is a fundamental skill in algebra and is essential for solving more complex problems.

Therefore, the simplified form of √400x¹⁰⁰ is 20x⁵⁰. This result is obtained by breaking down the expression into its numerical and variable components, simplifying each part separately, and then combining the simplified terms. The process involves recognizing perfect squares, applying the properties of radicals, and understanding the relationship between square roots and exponents. This example illustrates the power of simplification in mathematics, allowing us to represent complex expressions in a more manageable and understandable form. The answer, 20x⁵⁰, is not only the simplified form but also the most efficient and clear representation of the original expression. It highlights the importance of mastering simplification techniques, which are fundamental to success in algebra and higher-level mathematics. By following a systematic approach and understanding the underlying principles, anyone can confidently tackle radical simplification problems and arrive at the correct solution. This final answer serves as a testament to the effectiveness of the step-by-step approach we've employed, ensuring both accuracy and clarity in our solution.

In conclusion, the journey of simplifying √400x¹⁰⁰ has provided a valuable insight into the world of radical expressions and the techniques used to manipulate them. We've seen how breaking down a complex expression into simpler components, such as numerical coefficients and variable terms, can make the simplification process much more manageable. The recognition of perfect squares, both numerical and algebraic, is a key skill in this process. We've also reinforced the importance of understanding the properties of radicals and the relationship between square roots and exponents. The final simplified form, 20x⁵⁰, is a testament to the power of these techniques and the clarity they bring to mathematical expressions. This exercise not only provides a solution to a specific problem but also enhances our overall understanding of mathematical principles. The ability to simplify radical expressions is a fundamental skill that extends far beyond this example, playing a crucial role in various areas of mathematics and science. By mastering these techniques, we equip ourselves with the tools necessary to tackle more complex problems and deepen our appreciation for the elegance and precision of mathematics.

Answer: D. 20x⁵⁰