Equivalent Expression Of 10 Square Root 5 A Step-by-Step Solution
In the realm of mathematics, radicals, often represented by the square root symbol (√), play a pivotal role in expressing and manipulating numerical values. Among the various operations involving radicals, simplifying and identifying equivalent expressions stand out as essential skills. This article delves into the intricacies of simplifying radical expressions, with a specific focus on unraveling the equivalent form of 10√5. Our mission is to provide a comprehensive guide, ensuring clarity and understanding for students and enthusiasts alike. Prepare to embark on a journey through the world of radicals, where we'll dissect the problem, explore the underlying concepts, and ultimately arrive at the correct solution.
Understanding the Basics of Radicals: Before we plunge into the heart of the problem, let's lay a solid foundation by revisiting the fundamental principles of radicals. A radical expression consists of a radical symbol (√), a radicand (the number under the radical symbol), and an index (the small number placed above and to the left of the radical symbol, indicating the root to be extracted). In the case of square roots, the index is typically omitted, implying a root of 2. Simplifying a radical expression involves extracting any perfect square factors from the radicand, thereby expressing the radical in its most concise form. For example, √16 can be simplified to 4, since 16 is a perfect square (4 x 4). Similarly, √75 can be simplified to 5√3 by factoring 75 into 25 x 3, where 25 is a perfect square (5 x 5). These basic principles form the bedrock of our exploration into the equivalent expression of 10√5.
Deconstructing 10√5: Unveiling the Hidden Square Root: Now that we've refreshed our understanding of radical simplification, let's turn our attention to the expression at hand: 10√5. The key to finding an equivalent expression lies in recognizing that the coefficient 10 can be expressed as the square root of a perfect square. In other words, we can rewrite 10 as √100, since 10 x 10 equals 100. This seemingly simple transformation opens the door to simplifying the entire expression under a single radical symbol. By substituting √100 for 10 in the original expression, we get √100 * √5. The product of square roots rule allows us to combine these two radicals under a single radical symbol: √(100 * 5). This step is crucial, as it consolidates the expression and sets the stage for the final simplification. By multiplying 100 and 5, we arrive at √(500), which represents the equivalent expression of 10√5.
With a clear understanding of the simplification process, let's dissect the provided options and identify the correct equivalent expression for 10√5. Each option presents a radical expression, and our task is to determine which one matches the simplified form we derived earlier: √(500). This involves carefully examining each option, applying the principles of radical simplification, and comparing the result with our target expression. By systematically analyzing each choice, we'll not only pinpoint the correct answer but also reinforce our understanding of radical manipulation.
Option A: √500 – The Correct Match: The first option, √500, immediately catches our attention. It aligns perfectly with the simplified expression we derived earlier, √(500). This direct match strongly suggests that option A is the correct equivalent expression for 10√5. However, to ensure absolute certainty, let's briefly revisit our simplification steps. We started with 10√5, rewrote 10 as √100, multiplied the radicals to get √(100 * 5), and finally simplified to √(500). This meticulous process confirms that √500 is indeed the equivalent expression we sought. Therefore, option A stands as the unequivocal answer.
Option B: √105 – A Mismatch: Option B, √105, presents a different scenario. To assess its equivalence to 10√5, we need to determine if √105 can be simplified to match √(500). Factoring 105, we find its prime factors to be 3, 5, and 7. None of these factors appear in pairs, meaning that 105 does not have any perfect square factors. Consequently, √105 cannot be simplified further. This inability to simplify √105 to a form resembling √(500) definitively rules out option B as the correct answer. The mismatch in simplification potential highlights the importance of identifying perfect square factors when dealing with radical expressions.
Option C: √50 – A Closer Examination: Option C, √50, warrants a closer examination. At first glance, it might seem closer to √(500) than option B, but a thorough simplification is necessary to confirm its equivalence. Factoring 50, we find its prime factors to be 2, 5, and 5. The presence of a pair of 5s indicates a perfect square factor of 25 (5 x 5). We can therefore rewrite √50 as √(25 * 2), which simplifies to 5√2. Comparing this simplified form with 10√5, we observe a clear discrepancy. 5√2 is not equivalent to 10√5, as the coefficients and radicands differ. This detailed simplification process demonstrates that option C is not the correct equivalent expression.
Option D: √15 – A Clear Divergence: The final option, √15, presents the most straightforward case of divergence. Factoring 15, we find its prime factors to be 3 and 5. Neither of these factors appears in pairs, indicating the absence of perfect square factors. Consequently, √15 cannot be simplified any further. This lack of simplification potential, coupled with the distinct difference in numerical value, makes it evident that √15 is not equivalent to 10√5. Option D, therefore, is unequivocally ruled out as the correct answer.
Through our meticulous analysis of each option, we've systematically narrowed down the possibilities and arrived at a definitive conclusion. Option A, √500, stands as the sole equivalent expression for 10√5. Our journey began with a deconstruction of 10√5, followed by a step-by-step simplification process that yielded √(500). We then dissected each option, applying the principles of radical simplification and comparing the results with our target expression. This comprehensive approach not only led us to the correct answer but also reinforced our understanding of radical manipulation.
Simplifying radical expressions is more than just a mathematical exercise; it's a gateway to mathematical fluency. By mastering the concepts and techniques involved, students gain a deeper appreciation for the interconnectedness of mathematical ideas. Our exploration of 10√5 serves as a microcosm of this broader journey. The ability to rewrite coefficients as radicals, apply the product of square roots rule, and identify perfect square factors are all essential skills that extend beyond this specific problem. As students continue to hone these skills, they'll find themselves better equipped to tackle more complex mathematical challenges.
In conclusion, the expression equivalent to 10√5 is unequivocally √500. This determination stems from a systematic simplification process, a thorough analysis of the provided options, and a solid understanding of radical principles. Embrace the journey of mathematical exploration, and you'll unlock the beauty and power of these fundamental concepts.
