Simplify 5√24 A Step-by-Step Guide

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Simplifying radicals is a fundamental skill in algebra, and in this comprehensive guide, we'll tackle the problem of simplifying 5245 \sqrt{24}. This seemingly complex expression can be broken down into simpler terms, revealing its true value. We'll explore the process step-by-step, ensuring you grasp the underlying concepts and can confidently simplify similar radical expressions.

Understanding Radicals: The Foundation of Simplification

Before we dive into the specifics of simplifying 5245 \sqrt{24}, let's establish a solid understanding of what radicals represent. A radical, denoted by the symbol  {\sqrt{\ }}, indicates a root of a number. The most common type is the square root, which asks: "What number, when multiplied by itself, equals the number under the radical sign?" For instance, 9=3{\sqrt{9} = 3} because 3 multiplied by 3 equals 9. The number under the radical sign is called the radicand.

The process of simplifying radicals involves identifying perfect square factors within the radicand. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25). By extracting these perfect square factors, we can reduce the radical to its simplest form. This simplification is crucial for performing further operations with radicals, such as addition, subtraction, multiplication, and division.

In the expression 5245 \sqrt{24}, 5 is a coefficient, and 24{\sqrt{24}} is the radical part. Our goal is to simplify 24{\sqrt{24}} by finding its perfect square factors. Understanding this foundational concept is the key to mastering radical simplification.

Step-by-Step Simplification of 5245 \sqrt{24}

Now, let's embark on the journey of simplifying $5 \sqrt{24)}. We'll break down the process into manageable steps, ensuring clarity and comprehension every step of the way.

1. Identifying Perfect Square Factors

The first crucial step is to identify the perfect square factors of the radicand, which in this case is 24. A factor is a number that divides evenly into another number. We need to find factors of 24 that are also perfect squares. Let's list the factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24. Among these, 1 and 4 are perfect squares (1 = 1², 4 = 2²). We choose 4 because it's the largest perfect square factor of 24. This allows us to simplify the radical more efficiently.

2. Rewriting the Radicand

Once we've identified the perfect square factor (4), we rewrite the radicand (24) as a product of the perfect square and its remaining factor. In this case, we can express 24 as 4 * 6. So, 24{\sqrt{24}} becomes 4∗6{\sqrt{4 * 6}}. This step is essential because it allows us to apply the product property of radicals, which states that the square root of a product is equal to the product of the square roots.

3. Applying the Product Property of Radicals

The product property of radicals is a powerful tool for simplification. It states that a∗b=a∗b{\sqrt{a * b} = \sqrt{a} * \sqrt{b}}. Applying this property to our expression, 4∗6{\sqrt{4 * 6}} becomes 4∗6{\sqrt{4} * \sqrt{6}}. This separation allows us to simplify the perfect square root individually.

4. Simplifying the Perfect Square Root

Now we simplify 4{\sqrt{4}}. Since 4 is a perfect square (2² = 4), its square root is 2. Therefore, 4=2{\sqrt{4} = 2}. Our expression now looks like 2 * 6{\sqrt{6}}. The radical 6{\sqrt{6}} cannot be simplified further because 6 has no perfect square factors other than 1. This step highlights the importance of recognizing and extracting perfect square factors.

5. Reassembling the Expression

We've simplified 24{\sqrt{24}} to 2 6{\sqrt{6}}. Now we need to incorporate the original coefficient of 5. Remember, our original expression was 5 24{\sqrt{24}}. We substitute the simplified radical back into the expression: 5 * (2 6{\sqrt{6}}).

6. Final Simplification

The final step is to multiply the coefficients. We have 5 * 2 6{\sqrt{6}}, which simplifies to 10 6{\sqrt{6}}. This is the simplest form of the expression. We have successfully extracted the perfect square factor and simplified the radical.

The Simplified Result: 10610 \sqrt{6}

After meticulously following the steps, we've arrived at the simplified form of 5245 \sqrt{24}, which is 10610 \sqrt{6}. This result is not only more concise but also easier to work with in further mathematical operations. The process we've undertaken exemplifies the power of breaking down complex problems into smaller, manageable steps.

Importance of Simplifying Radicals

Simplifying radicals is not just an academic exercise; it has practical applications in various fields, including engineering, physics, and computer graphics. Simplified radicals make calculations easier, lead to more elegant solutions, and provide a deeper understanding of mathematical relationships. For example, in geometry, simplifying radicals is crucial for finding exact lengths and areas. In physics, it's essential for calculations involving energy and momentum. Mastering radical simplification empowers you to tackle real-world problems with greater confidence and efficiency.

Practice Makes Perfect: Examples and Exercises

To solidify your understanding of simplifying radicals, let's explore a few more examples and exercises. These examples will reinforce the steps we've covered and provide you with the opportunity to apply your knowledge in different contexts.

Example 1: Simplify 3503 \sqrt{50}

  1. Identify Perfect Square Factors: The factors of 50 are 1, 2, 5, 10, 25, and 50. The largest perfect square factor is 25.
  2. Rewrite the Radicand: 50=25∗2{\sqrt{50} = \sqrt{25 * 2}}
  3. Apply the Product Property: 25∗2=25∗2{\sqrt{25 * 2} = \sqrt{25} * \sqrt{2}}
  4. Simplify the Perfect Square Root: 25=5{\sqrt{25} = 5}
  5. Reassemble the Expression: 3 * (5 2{\sqrt{2}})
  6. Final Simplification: 15 2{\sqrt{2}}

Therefore, the simplified form of 3503 \sqrt{50} is 15215 \sqrt{2}.

Example 2: Simplify −272-2 \sqrt{72}

  1. Identify Perfect Square Factors: The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The largest perfect square factor is 36.
  2. Rewrite the Radicand: 72=36∗2{\sqrt{72} = \sqrt{36 * 2}}
  3. Apply the Product Property: 36∗2=36∗2{\sqrt{36 * 2} = \sqrt{36} * \sqrt{2}}
  4. Simplify the Perfect Square Root: 36=6{\sqrt{36} = 6}
  5. Reassemble the Expression: -2 * (6 2{\sqrt{2}})
  6. Final Simplification: -12 2{\sqrt{2}}

Thus, the simplified form of −272-2 \sqrt{72} is −122-12 \sqrt{2}.

Exercise 1: Simplify 4484 \sqrt{48}

Exercise 2: Simplify −598-5 \sqrt{98}

Exercise 3: Simplify 21252 \sqrt{125}

Try these exercises on your own, applying the steps we've discussed. Check your answers with the solutions provided below.

Common Mistakes to Avoid

While simplifying radicals, it's crucial to avoid common pitfalls that can lead to incorrect results. Here are some frequent mistakes to watch out for:

  • Failing to Identify the Largest Perfect Square Factor: Choosing a smaller perfect square factor will require further simplification steps. Always aim for the largest one to streamline the process.
  • Incorrectly Applying the Product Property: Ensure you are only separating the radicand into factors, not adding or subtracting terms.
  • Forgetting the Coefficient: Remember to multiply the coefficient outside the radical by the simplified radical.
  • Stopping Too Early: Make sure the radicand has no remaining perfect square factors before considering the radical fully simplified.

By being mindful of these common mistakes, you can improve your accuracy and efficiency in simplifying radicals.

Advanced Techniques and Applications

As you become more proficient in simplifying radicals, you can explore advanced techniques and applications. One such technique is simplifying radicals with variables. For example, x3{\sqrt{x^3}} can be simplified to xx{\sqrt{x}}. Understanding how to handle variables within radicals expands your problem-solving capabilities.

Furthermore, simplifying radicals is essential in various mathematical contexts, including solving quadratic equations, working with trigonometric functions, and performing calculus operations. The ability to simplify radicals efficiently will serve you well in higher-level mathematics courses and beyond.

Conclusion: Mastering Radical Simplification

In this comprehensive guide, we've explored the process of simplifying radicals, focusing on the example of 5245 \sqrt{24}. We've broken down the steps, provided examples, highlighted common mistakes, and discussed advanced techniques and applications. By mastering radical simplification, you'll gain a valuable skill that enhances your mathematical proficiency and opens doors to more complex problem-solving scenarios.

Remember, practice is key. The more you work with simplifying radicals, the more confident and adept you'll become. Embrace the challenge, and you'll find that simplifying radicals is not only a fundamental skill but also a rewarding endeavor in the world of mathematics.

Solutions to Exercises:

  • Exercise 1: 448=1634 \sqrt{48} = 16 \sqrt{3}
  • Exercise 2: −598=−352-5 \sqrt{98} = -35 \sqrt{2}
  • Exercise 3: 2125=1052 \sqrt{125} = 10 \sqrt{5}