Simplify Radicals Identifying The Expression Equivalent To 5√3

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In the world of mathematics, simplifying radicals is a fundamental skill, especially when dealing with expressions involving square roots. Understanding how to manipulate and simplify these expressions is crucial for success in algebra and beyond. In this comprehensive guide, we will delve into the intricacies of simplifying radicals, with a specific focus on the expression ${5\sqrt{3}}$. Our primary goal is to identify which of the given options—${\sqrt{30}}$, ${\sqrt{45}}$, ${\sqrt{75}}$, or ${\sqrt{15}}$—simplifies to ${5\sqrt{3}}$. By the end of this exploration, you will not only know the correct answer but also grasp the underlying principles that govern the simplification of radicals. This knowledge will empower you to tackle similar problems with confidence and precision.

To begin our journey, let's first establish a clear understanding of what it means to simplify a radical. A radical expression is considered simplified when the radicand (the number under the square root symbol) has no perfect square factors other than 1. In other words, we want to extract any perfect square factors from the radicand and place them outside the square root symbol. This process involves identifying factors that are perfect squares, such as 4, 9, 16, 25, and so on, and then using the property ${\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}$ to separate and simplify the radical. Now, let's apply this knowledge to the expression ${5\sqrt{3}}$. This expression is already in a simplified form, as the radicand, 3, has no perfect square factors other than 1. Our task is to determine which of the given options can be simplified to match this expression. This requires us to reverse the simplification process, effectively moving the 5 back under the square root symbol. Before we dive into the options, let's reinforce the concept of perfect squares and their role in simplifying radicals. A perfect square is a number that can be obtained by squaring an integer. For example, 4 is a perfect square because it is the result of squaring 2 (${2^2 = 4}$). Similarly, 9 is a perfect square because it is the result of squaring 3 (${3^2 = 9}$). Recognizing perfect squares is crucial for simplifying radicals, as it allows us to extract these factors from the radicand and simplify the expression. In the context of our problem, we will use the fact that 25 is a perfect square (${5^2 = 25}$) to manipulate the expression ${5\sqrt{3}}$.

Breaking Down the Options: A Step-by-Step Analysis

Now that we have a solid foundation in simplifying radicals, let's examine each of the given options to determine which one simplifies to ${5\sqrt{3}}$. We will approach this systematically, breaking down each option and applying the principles of radical simplification.

Option A: ${\sqrt{30}}$

First, consider option A, ${\sqrt{30}}$. To determine if this simplifies to ${5\sqrt{3}}$, we need to find the factors of 30. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Among these factors, we are looking for perfect squares. However, none of the factors of 30, apart from 1, are perfect squares. This means that ${\sqrt{30}}$ cannot be simplified further using the method of extracting perfect square factors. Therefore, ${\sqrt{30}}$ does not simplify to ${5\sqrt{3}}$. The prime factorization of 30 is ${2 \times 3 \times 5}$, which further confirms that there are no repeated factors that would form a perfect square. This option serves as a good example of a radical that is already in its simplest form.

Option B: ${\sqrt{45}}$

Next, let's analyze option B, ${\sqrt{45}}$. The factors of 45 are 1, 3, 5, 9, 15, and 45. Among these factors, we identify 9 as a perfect square (${3^2 = 9}$). This means we can rewrite ${\sqrt{45}}$ as ${\sqrt{9 \times 5}}$. Using the property ${\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}$, we can separate this into ${\sqrt{9} \cdot \sqrt{5}}$. Since ${\sqrt{9} = 3}$, the expression simplifies to ${3\sqrt{5}}$. Comparing this to ${5\sqrt{3}}$, we see that they are not the same. Therefore, ${\sqrt{45}}$ does not simplify to ${5\sqrt{3}}$. This option demonstrates the process of identifying and extracting perfect square factors, but it leads to a different simplified radical.

Option C: ${\sqrt{75}}$

Now, let's turn our attention to option C, ${\sqrt{75}}$. The factors of 75 are 1, 3, 5, 15, 25, and 75. Among these factors, we identify 25 as a perfect square (${5^2 = 25}$). This allows us to rewrite ${\sqrt{75}}$ as ${\sqrt{25 \times 3}}$. Applying the property ${\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}$, we separate this into ${\sqrt{25} \cdot \sqrt{3}}$. Since ${\sqrt{25} = 5}$, the expression simplifies to ${5\sqrt{3}}$. This matches the target expression, so ${\sqrt{75}}$ is indeed the correct answer. This option perfectly illustrates how identifying and extracting perfect square factors can lead to the desired simplified radical.

Option D: ${\sqrt{15}}$

Finally, let's consider option D, ${\sqrt{15}}$. The factors of 15 are 1, 3, 5, and 15. None of these factors, apart from 1, are perfect squares. This means that ${\sqrt{15}}$ cannot be simplified further by extracting perfect square factors. Therefore, ${\sqrt{15}}$ does not simplify to ${5\sqrt{3}}$. Similar to option A, this option represents a radical that is already in its simplest form.

The Correct Answer: ${\sqrt{75}}$

Through our step-by-step analysis, we have determined that option C, ${\sqrt{75}}$, is the expression that simplifies to ${5\sqrt{3}}$. This was achieved by identifying 25 as a perfect square factor of 75 and extracting it from the radical. The process involved rewriting ${\sqrt{75}}$ as ${\sqrt{25 \times 3}}$, separating it into ${\sqrt{25} \cdot \sqrt{3}}$, and finally simplifying it to ${5\sqrt{3}}$. This exercise highlights the importance of recognizing perfect squares and applying the properties of radicals to simplify expressions effectively. By systematically examining each option, we not only found the correct answer but also reinforced our understanding of radical simplification.

Solidifying Understanding: Additional Insights and Practice

To further solidify your understanding of simplifying radicals, let's explore some additional insights and practice exercises. Simplifying radicals is a skill that improves with practice, and the more you work with these expressions, the more comfortable and proficient you will become.

The Importance of Prime Factorization

One valuable technique for simplifying radicals is prime factorization. Prime factorization involves breaking down a number into its prime factors—factors that are only divisible by 1 and themselves. For example, the prime factorization of 75 is ${3 \times 5 \times 5}$, which can be written as ${3 \times 5^2}$. This makes it clear that 25 (${5^2}$) is a perfect square factor. Prime factorization can be particularly helpful when dealing with larger numbers, as it provides a systematic way to identify perfect square factors. By breaking down the radicand into its prime factors, you can easily spot repeated factors that form perfect squares.

Working Backwards: Converting to the Form ${a\sqrt{b}}$

In our problem, we needed to determine which radical simplified to ${5\sqrt{3}}$. This involved a process of working backwards, effectively moving the 5 back under the square root symbol. To do this, we squared the 5 (${5^2 = 25}$) and multiplied it by the radicand, 3, resulting in ${\sqrt{25 \times 3} = \sqrt{75}}$. This technique is useful for verifying simplified radicals and for comparing different radical expressions. It allows you to convert an expression in the form ${a\sqrt{b}}$ back to the form ${\sqrt{c}}$, making it easier to compare radicals with different coefficients and radicands.

Practice Exercises

To reinforce your skills, try simplifying the following radicals:

1. ${\sqrt{18}}$
2. ${\sqrt{32}}$
3. ${\sqrt{48}}$
4. ${\sqrt{80}}$
5. ${\sqrt{98}}$

For each radical, identify the perfect square factors, extract them from the radicand, and simplify the expression. Check your answers to ensure you have correctly applied the principles of radical simplification. These practice exercises will help you build confidence and fluency in simplifying radicals.

Conclusion Mastering Radical Simplification

In this comprehensive guide, we have explored the intricacies of simplifying radicals, with a specific focus on the expression ${5\sqrt{3}}$. We have learned how to identify perfect square factors, extract them from the radicand, and simplify radical expressions. Through a step-by-step analysis of the given options, we determined that ${\sqrt{75}}$ simplifies to ${5\sqrt{3}}$. We also discussed the importance of prime factorization and the technique of working backwards to convert expressions to different forms. By understanding these principles and practicing regularly, you can master the art of simplifying radicals and confidently tackle a wide range of mathematical problems. Remember, the key to success in mathematics is consistent practice and a solid understanding of fundamental concepts. Keep exploring, keep practicing, and you will continue to grow your mathematical abilities.

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Which expression simplifies to ${5\sqrt{3}}$? Let's embark on a journey through the realm of radical simplification, where we'll unravel the mystery behind this question. In this comprehensive guide, we'll dissect the art of simplifying radicals, equipping you with the tools to confidently tackle such problems. Our focus will be on identifying which of the given options simplifies to ${5\sqrt{3}}$. But before we dive into the specifics, let's lay a solid foundation by revisiting the fundamentals of radicals and their simplification.

At its core, simplifying a radical means expressing it in its most basic form, where the radicand (the number under the square root symbol) has no perfect square factors other than 1. Think of it as decluttering a room – we want to remove any unnecessary baggage and leave behind only the essentials. To achieve this, we need to identify perfect square factors within the radicand and extract their square roots. For instance, in ${\sqrt{16}}$, 16 is a perfect square (4 * 4), so we can simplify it to 4. But what about more complex cases? That's where our step-by-step approach comes in handy. We'll break down each option, meticulously examining its factors to pinpoint those perfect squares. This process not only leads us to the correct answer but also deepens our understanding of radical manipulation. So, buckle up, and let's embark on this adventure together!

Now, let's pivot to our specific challenge: pinpointing which expression simplifies to ${5\sqrt{3}}$. This requires a strategic approach. Instead of directly simplifying the options, we'll employ a clever tactic: working backward. We'll take ${5\sqrt{3}}$ and transform it into an equivalent expression with the 5 tucked neatly under the radical sign. This reverse engineering provides a clear target for our simplification efforts. Remember, when we move a number under the square root, we must square it first. This is because the square root is the inverse operation of squaring. So, to bring 5 under the radical, we square it (5 * 5 = 25) and multiply it by the existing radicand, 3. This gives us ${\sqrt{25 * 3}}$, which simplifies to ${\sqrt{75}}$. Ah, ${\sqrt{75}}$ becomes our beacon, guiding us through the options. Now, our mission is clear: identify the option that, when simplified, unveils ${\sqrt{75}}$. This methodical approach not only streamlines the problem-solving process but also illuminates the underlying principles of radical manipulation. We're not just seeking the answer; we're mastering the art of simplification.

Dissecting the Options: A Quest for the Perfect Match

With our target, ${\sqrt{75}}$, firmly in sight, let's embark on a quest to dissect each option and determine its simplified form. We'll approach each radical methodically, breaking down the radicand into its prime factors and identifying any perfect square partners lurking within.

Option A: The Enigma of ${\sqrt{30}}$

First, let's confront the enigma of ${\sqrt{30}}$. To unravel its secrets, we'll embark on a factor-finding mission. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. As we scrutinize this list, we search for perfect squares – numbers that are the product of an integer multiplied by itself. Alas, 30 yields no perfect square partners, save for the ever-present 1. This indicates that ${\sqrt{30}}$ is already in its simplest form, a lone wolf that cannot be tamed further. Its prime factorization (2 * 3 * 5) confirms this, revealing no repeated factors that could form a perfect square. Thus, ${\sqrt{30}}$ remains steadfast in its original form, failing to match our target of ${\sqrt{75}}$. This option serves as a valuable reminder that not all radicals can be simplified; sometimes, the radicand is already in its most basic state.

Option B: Deciphering ${\sqrt{45}}$

Next, we turn our attention to ${\sqrt{45}}$, determined to decipher its hidden form. The factors of 45 beckon us: 1, 3, 5, 9, 15, and 45. Eureka! We spot a familiar face – 9, a perfect square (3 * 3). This is our key to simplification. We can rewrite ${\sqrt{45}}$ as ${\sqrt{9 * 5}}$. Now, the magic happens. We invoke the property of radicals that allows us to separate the product under the square root: ${\sqrt{9 * 5} = \sqrt{9} * \sqrt{5}}$. Since ${\sqrt{9}}$ gracefully transforms into 3, we arrive at ${3\sqrt{5}}$. While ${3\sqrt{5}}$ is a simplified form, it doesn't quite align with our target, ${\sqrt{75}}$. Option B demonstrates the power of identifying perfect square factors and extracting their square roots, but it leads us down a different path. It highlights the importance of not just simplifying but simplifying towards the specific form we seek.

Option C: Unveiling ${\sqrt{75}}$'s True Identity

Now, we confront the moment of truth: ${\sqrt{75}}$. This is where our target expression resides, but can we unveil its true identity through simplification? Let's delve into the factors of 75: 1, 3, 5, 15, 25, and 75. Our eyes immediately lock onto 25, a perfect square (5 * 5). This is the golden ticket we've been searching for. We rewrite ${\sqrt{75}}$ as ${\sqrt{25 * 3}}$. Following the same dance as before, we separate the radicals: ${\sqrt{25 * 3} = \sqrt{25} * \sqrt{3}}$. ${\sqrt{25}}$ gracefully transforms into 5, leaving us with the triumphant result: ${5\sqrt{3}}$. This is our target! ${\sqrt{75}}$ indeed simplifies to ${5\sqrt{3}}$, making it the correct answer. Option C not only solves the problem but also validates our initial strategy of working backward, solidifying our understanding of the simplification process.

Option D: The Simplicity of ${\sqrt{15}}$

Finally, we arrive at ${\sqrt{15}}$. Let's give it the same scrutiny as its predecessors. The factors of 15 are 1, 3, 5, and 15. A quick scan reveals no perfect squares lurking within. This means ${\sqrt{15}}$ is already in its simplest form, a testament to its inherent simplicity. Its prime factorization (3 * 5) confirms this, with no repeated factors to exploit. Thus, ${\sqrt{15}}$ stands alone, failing to match our target of ${\sqrt{75}}$. Option D serves as a final reminder that not all radicals can be simplified further, and recognizing these cases is just as crucial as mastering the simplification process itself.

Conclusion Embracing the Art of Radical Simplification

Our quest concludes, and the answer shines brightly: Option C, ${\sqrt{75}}$, is the expression that simplifies to ${5\sqrt{3}}$. We arrived at this triumphant conclusion through a methodical journey, starting with a strategic decision to work backward, transforming our target expression into ${\sqrt{75}}$. We then dissected each option, meticulously examining its factors and extracting any perfect squares lurking within. This process not only led us to the correct answer but also deepened our understanding of radical simplification. We've learned the importance of identifying perfect square factors, the power of prime factorization, and the elegance of working strategically. But most importantly, we've embraced the art of radical simplification, a skill that empowers us to confidently navigate the world of mathematical expressions. So, let this be not the end, but the beginning of your journey into the fascinating realm of radicals! Keep exploring, keep simplifying, and keep embracing the beauty of mathematics.

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In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. This guide is dedicated to unraveling the complexities of radical simplification, focusing on the specific problem of identifying which expression is equivalent to ${5\sqrt{3}}$. We will explore the concept of radicals, delve into the methods of simplification, and step-by-step analyze each option to determine the correct answer. Whether you are a student seeking to enhance your understanding or a math enthusiast eager to expand your knowledge, this comprehensive guide will provide you with the necessary tools and insights.

To begin our journey, it's crucial to grasp the essence of radicals. A radical is a mathematical expression that uses a root, such as a square root, cube root, or nth root. The most common type of radical is the square root, denoted by the symbol ${\sqrt{ }}$. The number inside the radical symbol is called the radicand. Simplifying radicals involves expressing them in their simplest form, where the radicand has no perfect square factors other than 1. This process often requires identifying perfect square factors within the radicand and extracting their square roots. For instance, ${\sqrt{16}}$ can be simplified to 4 because 16 is a perfect square (4 * 4 = 16). Understanding this basic principle is the cornerstone of simplifying more complex radical expressions. To master radical simplification, it is also essential to be familiar with the properties of radicals, such as the product property and the quotient property. The product property states that ${\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}$, which allows us to separate the factors within a radical. The quotient property states that ${\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}}$, enabling us to simplify radicals involving fractions. These properties serve as powerful tools in our quest to simplify radicals efficiently and accurately. As we proceed, we will apply these principles to the given options, carefully dissecting each expression to reveal its simplest form and ultimately identify the one that matches ${5\sqrt{3}}$.

Our primary goal is to determine which of the given options simplifies to ${5\sqrt{3}}$. This requires a strategic approach. Instead of directly simplifying each option, we can employ a technique of working backward. We begin by manipulating the expression ${5\sqrt{3}}$ to a form that allows us to easily compare it with the options. The key is to move the coefficient 5 inside the square root. Remember, when we move a number inside a square root, we must square it first. This is because the square root operation is the inverse of squaring. Therefore, we square 5, which gives us 25, and then multiply it by the radicand 3. This results in ${\sqrt{25 \cdot 3}}$, which simplifies to ${\sqrt{75}}$. Now, our task is clear: we need to identify which of the given options simplifies to ${\sqrt{75}}$. This backward approach transforms our problem into a more manageable one, providing us with a clear target for our simplification efforts. By establishing ${\sqrt{75}}$ as our benchmark, we can systematically evaluate each option and determine its simplified form. This method not only streamlines the problem-solving process but also reinforces our understanding of the relationship between coefficients and radicands in radical expressions. With our target in mind, let's proceed to analyze each option and unveil its true identity.

Step-by-Step Analysis of the Options: Finding the Perfect Match

With ${\sqrt{75}}$ as our target, we will now embark on a detailed analysis of each option. Our approach involves simplifying each radical expression and comparing the result with our target. This step-by-step process will not only lead us to the correct answer but also enhance our understanding of radical simplification techniques.

Option A: Unveiling the Mystery of ${\sqrt{30}}$

Our first task is to simplify ${\sqrt{30}}$. To do this, we need to identify the factors of 30. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. We are looking for perfect square factors, which are numbers that can be expressed as the square of an integer (e.g., 4, 9, 16, 25). Upon inspection, we find that none of the factors of 30, other than 1, are perfect squares. This indicates that ${\sqrt{30}}$ cannot be simplified further. Its prime factorization (2 * 3 * 5) also confirms this, as there are no repeated prime factors that would form a perfect square. Therefore, ${\sqrt{30}}$ is already in its simplest form and does not match our target of ${\sqrt{75}}$. This option serves as a reminder that not all radicals can be simplified, and recognizing this is an essential skill in working with radicals.

Option B: Exploring the Depths of ${\sqrt{45}}$

Next, let's delve into the simplification of ${\sqrt{45}}$. We begin by listing the factors of 45: 1, 3, 5, 9, 15, and 45. Among these factors, we identify 9 as a perfect square (3 * 3 = 9). This is our key to simplification. We can rewrite ${\sqrt{45}}$ as ${\sqrt{9 \cdot 5}}$. Using the product property of radicals, we can separate this into ${\sqrt{9} \cdot \sqrt{5}}$. Since ${\sqrt{9}}$ simplifies to 3, the expression becomes ${3\sqrt{5}}$. While this is a simplified form, it does not match our target of ${\sqrt{75}}$. Therefore, ${\sqrt{45}}$ is not the expression we are looking for. This option demonstrates the process of identifying and extracting perfect square factors, but it leads us to a different simplified radical.

Option C: Unmasking the True Identity of ${\sqrt{75}}$

Now, we turn our attention to ${\sqrt{75}}$, our target expression. To simplify it, we first identify the factors of 75: 1, 3, 5, 15, 25, and 75. We immediately notice that 25 is a perfect square (5 * 5 = 25). This allows us to rewrite ${\sqrt{75}}$ as ${\sqrt{25 \cdot 3}}$. Applying the product property of radicals, we separate this into ${\sqrt{25} \cdot \sqrt{3}}$. Since ${\sqrt{25}}$ simplifies to 5, the expression becomes ${5\sqrt{3}}$. This is precisely the expression we were seeking! Therefore, ${\sqrt{75}}$ simplifies to ${5\sqrt{3}}$, making it the correct answer. This option validates our strategic approach of working backward and confirms our understanding of radical simplification techniques.

Option D: Examining the Simplicity of ${\sqrt{15}}$

Finally, let's examine ${\sqrt{15}}$. To simplify it, we list the factors of 15: 1, 3, 5, and 15. We observe that none of these factors, other than 1, are perfect squares. This indicates that ${\sqrt{15}}$ is already in its simplest form. Its prime factorization (3 * 5) also confirms this, as there are no repeated prime factors to form a perfect square. Therefore, ${\sqrt{15}}$ does not simplify to ${5\sqrt{3}}$. Similar to option A, this option serves as a reminder that some radicals are already in their simplest form and cannot be simplified further.

Conclusion Mastering the Art of Radical Simplification

Through our detailed analysis, we have successfully identified that option C, ${\sqrt{75}}$, is the expression that simplifies to ${5\sqrt{3}}$. This was achieved by systematically examining each option, identifying perfect square factors, and applying the properties of radicals. Our strategic approach of working backward, starting with the target expression and transforming it into a comparable form, proved to be highly effective. We also reinforced the importance of recognizing when a radical is already in its simplest form. This comprehensive guide has provided you with the necessary tools and insights to master the art of radical simplification. Remember, the key to success in mathematics lies in understanding the fundamental principles and practicing regularly. As you continue your mathematical journey, embrace the challenges and celebrate the triumphs, and you will undoubtedly achieve your goals.