Simplifying Cube Root Product Of 24 And 45 A Detailed Solution

by Admin 63 views

Introduction: Mastering Cube Root Multiplication

In the realm of mathematics, simplifying expressions involving radicals, particularly cube roots, is a fundamental skill. This article delves into the process of simplifying the product 243â‹…453{\sqrt[3]{24} \cdot \sqrt[3]{45}}, guiding you through the steps to arrive at the correct answer. We will not only identify the solution but also explore the underlying principles of cube root manipulation, ensuring a comprehensive understanding of the topic. Understanding cube roots is crucial for simplifying radical expressions, and this exploration will equip you with the necessary tools and techniques. Let's embark on this mathematical journey together, unraveling the complexities of cube roots and their properties.

The expression we aim to simplify, 243â‹…453{\sqrt[3]{24} \cdot \sqrt[3]{45}}, involves the multiplication of two cube roots. The key to simplifying such expressions lies in understanding the properties of radicals and prime factorization. We will break down each number under the cube root into its prime factors, looking for perfect cubes that can be extracted. This process allows us to reduce the expression to its simplest form. The simplification of cube roots often involves identifying perfect cube factors within the radicand. By extracting these factors, we can express the radical in a more manageable form. This technique is widely used in algebra and calculus to simplify complex expressions and solve equations. Mastering this skill is essential for anyone pursuing further studies in mathematics. The journey of simplifying this expression will not only provide the solution but also enhance your problem-solving abilities in mathematics. Remember, practice is the key to mastering any mathematical concept. By working through examples like this, you'll develop a deeper understanding of cube roots and their properties.

The ability to manipulate and simplify radical expressions is a cornerstone of algebraic proficiency. This particular problem highlights the importance of prime factorization and the product rule for radicals. We'll see how breaking down the numbers 24 and 45 into their prime factors allows us to identify perfect cube factors, which can then be extracted from the cube roots. This process not only simplifies the expression but also reveals the underlying mathematical structure. The product rule for radicals, which states that anâ‹…bn=abn{\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}}, is a fundamental tool in simplifying expressions like this. By applying this rule, we can combine the two cube roots into a single cube root, making the simplification process more straightforward. This example serves as a reminder of the interconnectedness of mathematical concepts. Prime factorization, radical properties, and simplification techniques all come together to solve this problem. By understanding these connections, you can develop a more holistic approach to mathematics and problem-solving.

Step-by-Step Solution: Deconstructing the Cube Roots

To simplify the product 243â‹…453{\sqrt[3]{24} \cdot \sqrt[3]{45}}, we begin by applying the property of radicals that allows us to combine the two cube roots into one: a3â‹…b3=aimesb3{\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a imes b}}. Thus, our expression becomes 24imes453{\sqrt[3]{24 imes 45}}. Now, we need to find the product of 24 and 45, which is 1080. So, we have 10803{\sqrt[3]{1080}}. The next crucial step involves prime factorization, breaking down 1080 into its prime factors. This will help us identify any perfect cubes within the radicand. By expressing 1080 as a product of its prime factors, we can easily spot the cube roots that can be extracted. This is a standard technique for simplifying radicals and is essential for solving a wide range of mathematical problems. Prime factorization not only helps in simplifying radicals but also in finding the greatest common divisor (GCD) and the least common multiple (LCM) of numbers. It is a fundamental concept in number theory and has numerous applications in mathematics and computer science.

Let's perform the prime factorization of 1080. We can start by dividing 1080 by the smallest prime number, 2. 1080 divided by 2 is 540. We can divide 540 by 2 again, resulting in 270. Dividing 270 by 2 gives us 135. Now, 135 is not divisible by 2, so we move to the next prime number, 3. 135 divided by 3 is 45. Dividing 45 by 3 gives us 15, and dividing 15 by 3 gives us 5. Finally, 5 is a prime number. Therefore, the prime factorization of 1080 is 23imes33imes5{2^3 imes 3^3 imes 5}. Now we can rewrite the cube root expression as 23imes33imes53{\sqrt[3]{2^3 imes 3^3 imes 5}}. We can see that we have perfect cubes, namely 23{2^3} and 33{3^3}. The identification of perfect cubes within the prime factorization is the key to simplifying the cube root. These perfect cubes can be extracted from the radical, leaving us with a simplified expression. Recognizing perfect cubes and other perfect powers is a valuable skill in mathematics. It allows us to simplify expressions, solve equations, and gain a deeper understanding of number theory.

Using the property of radicals that aimesbn=animesbn{\sqrt[n]{a imes b} = \sqrt[n]{a} imes \sqrt[n]{b}}, we can separate the cube root of the product into the product of cube roots: 23imes33imes53=233imes333imes53{\sqrt[3]{2^3 imes 3^3 imes 5} = \sqrt[3]{2^3} imes \sqrt[3]{3^3} imes \sqrt[3]{5}}. The cube root of 23{2^3} is 2, and the cube root of 33{3^3} is 3. So, we have 2imes3imes53{2 imes 3 imes \sqrt[3]{5}}, which simplifies to 653{6\sqrt[3]{5}}. Therefore, the simplified form of the original expression 243imes453{\sqrt[3]{24} imes \sqrt[3]{45}} is 653{6\sqrt[3]{5}}. This process demonstrates the power of radical properties in simplifying complex expressions. By understanding and applying these properties, we can transform seemingly complicated problems into manageable steps. The ability to manipulate radicals is crucial in many areas of mathematics, including algebra, calculus, and trigonometry. This example showcases the importance of mastering these fundamental concepts.

Identifying the Correct Answer: Matching the Simplified Form

Now that we have simplified the expression 243â‹…453{\sqrt[3]{24} \cdot \sqrt[3]{45}} to 653{6\sqrt[3]{5}}, we can compare this result with the given options to identify the correct answer. The options provided are:

A. 693{\sqrt[3]{69}} B. 4(63){4(\sqrt[3]{6})} C. 6(53){6(\sqrt[3]{5})} D. 6(103){6(\sqrt[3]{10})}

By direct comparison, we can see that our simplified answer, 653{6\sqrt[3]{5}}, matches option C. Therefore, the correct answer is C. This step emphasizes the importance of accurate simplification and careful comparison. Even if the simplification process is correct, a mistake in the final comparison can lead to an incorrect answer. Therefore, it's crucial to double-check the final result and ensure it matches one of the given options. This meticulous approach is essential for success in mathematics and problem-solving in general. By carefully reviewing each step and verifying the final answer, we can minimize errors and build confidence in our solutions.

The process of verifying the answer is a critical step in problem-solving. It not only ensures accuracy but also reinforces the understanding of the concepts involved. In this case, by comparing our simplified result with the given options, we confirmed that option C is the correct answer. This process can also involve substituting the answer back into the original equation or expression to see if it holds true. This is a powerful technique for checking the validity of the solution and identifying any potential errors. Verification is an integral part of the mathematical process and should not be overlooked. It helps to develop a sense of confidence in the solution and provides valuable feedback on the problem-solving approach.

Conclusion: The Power of Simplification and Prime Factorization

In conclusion, we have successfully simplified the product 243â‹…453{\sqrt[3]{24} \cdot \sqrt[3]{45}} to 653{6\sqrt[3]{5}} by utilizing the properties of radicals and prime factorization. This exercise highlights the importance of understanding these fundamental mathematical concepts. The ability to simplify radical expressions is a valuable skill in various areas of mathematics, and this example provides a clear illustration of the techniques involved. By breaking down the problem into smaller, manageable steps, we were able to arrive at the correct solution. This step-by-step approach is crucial for tackling complex mathematical problems and building confidence in problem-solving abilities. The process of simplification not only provides the answer but also enhances our understanding of the underlying mathematical principles. This deeper understanding is essential for applying these concepts to other problems and situations.

This problem serves as a reminder of the interconnectedness of mathematical concepts. Prime factorization, radical properties, and simplification techniques all play a crucial role in solving this problem. By mastering these concepts, we can approach a wide range of mathematical challenges with confidence. The journey of simplifying this expression has not only provided the solution but also reinforced the importance of mathematical fundamentals. These fundamentals are the building blocks of more advanced concepts, and a strong foundation in these areas is essential for success in mathematics. By continuously practicing and applying these concepts, we can develop a deeper understanding and appreciation for the beauty and power of mathematics. Remember, mathematics is not just about finding the right answer; it's about understanding the process and developing critical thinking skills.

By mastering the techniques demonstrated in this example, you'll be well-equipped to tackle similar problems involving cube roots and other radical expressions. The key takeaways from this exploration are the importance of prime factorization, the properties of radicals, and the ability to break down complex expressions into simpler forms. These skills are not only valuable in mathematics but also in other fields that require analytical thinking and problem-solving abilities. So, continue practicing and exploring the world of mathematics, and you'll discover its endless possibilities and applications. The application of mathematical concepts extends far beyond the classroom. It is used in various fields, including science, engineering, finance, and computer science. By developing a strong foundation in mathematics, you'll open doors to a wide range of opportunities and careers. So, embrace the challenge of learning mathematics and enjoy the journey of discovery.