Simplifying Radicals Question 4 Solving $4 \sqrt{13}$

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In the realm of mathematics, simplifying radical expressions is a fundamental skill. It allows us to express numbers in their most concise and understandable form. Question 4 of 10 presents us with a compelling challenge: to identify which expression simplifies to $4\sqrt{13}$. This seemingly straightforward question opens the door to a fascinating exploration of square roots, perfect squares, and the art of simplification. Let's embark on this mathematical journey together, unraveling the intricacies of radical expressions and discovering the correct answer.

Understanding the Core Concepts: Radicals and Simplification

Before we delve into the specific options, it's crucial to lay a solid foundation by understanding the core concepts involved. A radical expression is essentially an expression that contains a root, most commonly a square root. The square root of a number is a value that, when multiplied by itself, gives you the original number. For instance, the square root of 9 is 3 because 3 * 3 = 9. The symbol for the square root is \√.

Simplifying a radical expression involves reducing it to its simplest form. This typically means removing any perfect square factors from the radicand (the number under the radical sign). A perfect square is a number that can be obtained by squaring an integer. Examples of perfect squares include 4 (2 * 2), 9 (3 * 3), 16 (4 * 4), and so on. The key to simplifying radicals lies in identifying these perfect square factors and extracting their square roots.

To illustrate this, consider the expression \√20. We can break down 20 into its factors: 20 = 4 * 5. Notice that 4 is a perfect square (2 * 2). Therefore, we can rewrite \√20 as \√(4 * 5). Using the property that \√(a * b) = \√a * \√b, we can further simplify this as \√4 * \√5. Since \√4 = 2, the simplified form of \√20 is 2\√5. This process of identifying perfect square factors and extracting their roots is the cornerstone of radical simplification.

Breaking Down $4\sqrt{13}$: The Target Expression

Our target expression, $4\sqrt{13}$, provides a crucial benchmark. It represents the simplified form we are aiming for. To effectively identify the equivalent expression from the given options, we need to understand how this expression is constructed. The number 4 outside the radical sign indicates that a perfect square has already been extracted. The \√13 signifies that 13 is the remaining radicand after the simplification process.

To reverse this process, we need to bring the 4 back under the radical sign. Remember that 4 is the square root of 16 (4 * 4 = 16). Therefore, we can rewrite 4 as \√16. Now, we can combine the two radicals: $4\sqrt{13} = \sqrt{16} * \sqrt{13}$. Using the property \√(a * b) = \√a * \√b in reverse, we get $4\sqrt{13} = \sqrt{16 * 13}$. Multiplying 16 and 13 gives us 208. Thus, $4\sqrt{13} = \sqrt{208}$. This transformation provides us with a crucial insight: the expression we seek must be equivalent to \√208 before simplification.

Evaluating the Options: A Step-by-Step Analysis

Now that we have a clear understanding of our target expression and the process of simplification, let's examine each option provided and determine if it simplifies to $4\sqrt{13}$.

  1. \sqrt{52}$: This is the first option we need to analyze. To simplify \√52, we need to identify its perfect square factors. The factors of 52 are 1, 2, 4, 13, 26, and 52. Among these, 4 is a perfect square (2 * 2). Therefore, we can rewrite \√52 as \√(4 * 13). Applying the property \√(a * b) = \√a * \√b, we get \√4 * \√13. Since \√4 = 2, the simplified form of \√52 is 2\√13. Comparing this with our target expression, $4\sqrt{13}$, we see that \√52 is not the correct answer. While it contains the \√13 term, the coefficient is 2, not 4.

  2. \sqrt{17}$: This option presents a different scenario. 17 is a prime number, meaning its only factors are 1 and itself. Consequently, it has no perfect square factors other than 1. Therefore, \√17 is already in its simplest form and cannot be simplified further. It is clear that \√17 is not equivalent to $4\sqrt{13}$, as it lacks both the coefficient 4 and the \√13 term.

  3. \sqrt{208}$: This option holds significant promise. As we deduced earlier, $4\sqrt{13}$ is equivalent to \√208 before simplification. To confirm this, let's simplify \√208. The factors of 208 are 1, 2, 4, 8, 13, 16, 26, 52, 104, and 208. Among these, 16 is a perfect square (4 * 4). We can rewrite \√208 as \√(16 * 13). Applying the property \√(a * b) = \√a * \√b, we get \√16 * \√13. Since \√16 = 4, the simplified form of \√208 is indeed $4\sqrt{13}$. This confirms that \√208 is the correct answer.

  4. \sqrt{29}$: Similar to \√17, 29 is a prime number. Its only factors are 1 and itself, meaning it has no perfect square factors other than 1. Therefore, \√29 is already in its simplest form. It is evident that \√29 is not equivalent to $4\sqrt{13}$, as it lacks both the coefficient 4 and the \√13 term.

The Verdict: Unveiling the Correct Answer

Through our step-by-step analysis, we have meticulously evaluated each option and identified the expression that simplifies to $4\sqrt{13}$. The correct answer is \√208. This option, when simplified by extracting the perfect square factor of 16, yields the target expression, $4\sqrt{13}$. The other options, \√52, \√17, and \√29, do not simplify to $4\sqrt{13}$ due to either a different coefficient or the absence of perfect square factors.

Key Takeaways: Mastering Radical Simplification

This exercise in identifying the expression equivalent to $4\sqrt{13}$ has provided us with valuable insights into the world of radical simplification. We have reinforced the importance of understanding the core concepts of radicals, perfect squares, and the properties of square roots. By systematically analyzing each option and applying the principles of simplification, we successfully pinpointed the correct answer.

The key takeaways from this exploration are:

  • Understanding the Definition of Square Roots: A square root of a number is a value that, when multiplied by itself, gives the original number.
  • Identifying Perfect Square Factors: Perfect squares are crucial for simplifying radicals. Recognizing perfect square factors within the radicand is the first step towards simplification.
  • Applying the Property \√(a * b) = \√a * \√b: This property allows us to separate radicals and extract the square roots of perfect square factors.
  • Simplifying Completely: Ensure that the radicand has no remaining perfect square factors after simplification.
  • Prime Numbers and Simplification: Radicals containing prime numbers (like 17 and 29 in our example) are already in their simplest form.

By mastering these concepts and techniques, you can confidently navigate the realm of radical expressions and simplify them with ease.

Conclusion: The Power of Mathematical Deduction

Question 4 of 10, while seemingly simple, has served as a powerful illustration of mathematical deduction. By carefully analyzing the target expression, understanding the principles of radical simplification, and systematically evaluating each option, we arrived at the correct answer: \√208. This process highlights the beauty of mathematics – the ability to break down complex problems into smaller, manageable steps and arrive at a logical solution.

Simplifying radical expressions is not just a mathematical exercise; it is a skill that enhances our understanding of numbers and their relationships. It empowers us to express quantities in their most concise and meaningful form. As we continue our mathematical journey, the ability to simplify radicals will prove invaluable in tackling more advanced concepts and challenges. So, embrace the power of simplification, and let it guide you towards mathematical mastery.