Simplifying Cube Roots √[3]{24} ⋅ √[3]{45} A Step-by-Step Guide
In the realm of mathematics, we often encounter expressions that seem daunting at first glance. Among these are radical expressions, particularly those involving cube roots. When faced with such expressions, it's crucial to possess the skills and knowledge to simplify them effectively. This article will delve into the process of simplifying radical expressions, specifically focusing on the product of cube roots. We'll unravel the mystery behind the expression √[3]{24} ⋅ √[3]{45}, breaking it down step by step to arrive at the correct answer. By the end of this journey, you'll not only understand the solution but also gain a deeper appreciation for the elegance and power of mathematical simplification.
The question we aim to address is: What is the product of √[3]{24} and √[3]{45}? The options presented are:
- A. √[3]{69}
- B. 4(√[3]{6})
- C. 6(√[3]{5})
- D. 6(√[3]{10})
To embark on this mathematical quest, we'll first lay the groundwork by understanding the fundamental concepts of cube roots and radical expressions. This will equip us with the necessary tools to tackle the problem at hand. Next, we'll embark on a step-by-step simplification process, carefully dissecting the expression and revealing its hidden simplicity. Along the way, we'll explore key strategies and techniques that can be applied to a wide range of radical simplification problems. Finally, we'll arrive at the solution and discuss its significance within the broader context of mathematics. So, let's begin our journey into the world of cube roots and radical expressions.
Understanding Cube Roots and Radical Expressions
Before we dive into the specific problem, it's essential to have a solid grasp of what cube roots and radical expressions are. A cube root of a number is a value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2 because 2 x 2 x 2 = 8. We denote the cube root of a number 'x' as √[3]{x}.
A radical expression is any mathematical expression that contains a radical symbol (√). The radical symbol indicates the root of a number, such as the square root (√), cube root (√[3]), or any higher-order root. The number under the radical symbol is called the radicand.
When dealing with radical expressions, it's often necessary to simplify them. Simplification involves rewriting the expression in a more concise and manageable form while preserving its value. One common technique for simplifying radical expressions is to factor the radicand and look for perfect cube factors. A perfect cube is a number that can be obtained by cubing an integer. For instance, 8 is a perfect cube because it's 2 cubed (2^3 = 8), and 27 is a perfect cube because it's 3 cubed (3^3 = 27).
The product of cube roots can be simplified using the property: √[3]{a} ⋅ √[3]{b} = √[3]{a ⋅ b}. This property states that the product of two cube roots is equal to the cube root of the product of their radicands. This property will be instrumental in simplifying the given expression. Understanding these fundamental concepts is crucial for successfully simplifying the expression √[3]{24} ⋅ √[3]{45}. Now, let's move on to the step-by-step simplification process.
Step-by-Step Simplification of √[3]{24} ⋅ √[3]{45}
Now that we have a firm understanding of cube roots and radical expressions, let's tackle the problem at hand: simplifying the product √[3]{24} ⋅ √[3]{45}. We'll break down the process into manageable steps, making it easier to follow and understand.
Step 1: Combine the Radicands
Using the property √[3]{a} ⋅ √[3]{b} = √[3]{a ⋅ b}, we can combine the two cube roots into a single cube root:
√[3]{24} ⋅ √[3]{45} = √[3]{24 ⋅ 45}
This step simplifies the expression by reducing two cube roots into one.
Step 2: Calculate the Product of the Radicands
Next, we multiply the radicands, 24 and 45:
24 ⋅ 45 = 1080
So, our expression now becomes:
√[3]{1080}
Step 3: Factor the Radicand
To simplify the cube root, we need to find the prime factorization of 1080. This involves breaking down 1080 into its prime factors:
1080 = 2 x 540
540 = 2 x 270
270 = 2 x 135
135 = 3 x 45
45 = 3 x 15
15 = 3 x 5
So, the prime factorization of 1080 is 2 x 2 x 2 x 3 x 3 x 3 x 5, which can be written as 2^3 x 3^3 x 5.
Step 4: Identify Perfect Cube Factors
Now, we look for perfect cube factors within the prime factorization. We have 2^3 (which is 8) and 3^3 (which is 27) as perfect cubes.
Step 5: Rewrite the Cube Root
We can rewrite the cube root as:
√[3]{1080} = √[3]{2^3 x 3^3 x 5}
Step 6: Simplify using the Cube Root Property
Using the property √[3]{a ⋅ b} = √[3]{a} ⋅ √[3]{b}, we can separate the cube root:
√[3]{2^3 x 3^3 x 5} = √[3]{2^3} ⋅ √[3]{3^3} ⋅ √[3]{5}
Now, we can take the cube roots of the perfect cubes:
√[3]{2^3} = 2
√[3]{3^3} = 3
So, the expression simplifies to:
2 ⋅ 3 ⋅ √[3]{5}
Step 7: Final Simplification
Finally, we multiply the constants:
2 ⋅ 3 ⋅ √[3]{5} = 6√[3]{5}
Therefore, the simplified form of √[3]{24} ⋅ √[3]{45} is 6√[3]{5}.
Solution and Its Significance
After carefully simplifying the expression √[3]24} ⋅ √[3]{45}, we have arrived at the solution**. This corresponds to option C in the given choices.
This exercise highlights the importance of understanding the properties of radicals and the techniques for simplifying them. By breaking down the problem into smaller, manageable steps, we were able to systematically unravel the complexity of the expression and arrive at a concise and elegant solution. The process involved combining radicands, factoring, identifying perfect cube factors, and applying the properties of cube roots.
The significance of this simplification extends beyond just finding the answer to a specific problem. It demonstrates the power of mathematical manipulation and the ability to transform seemingly complex expressions into simpler forms. This skill is invaluable in various areas of mathematics, including algebra, calculus, and beyond. Moreover, the process of simplification often reveals underlying patterns and relationships that might not be apparent in the original expression.
In conclusion, by mastering the techniques for simplifying radical expressions, we gain a deeper understanding of mathematical concepts and develop valuable problem-solving skills. The solution 6√[3]{5} is not just an answer; it's a testament to the elegance and power of mathematical simplification.
Additional Tips and Strategies for Simplifying Radical Expressions
Simplifying radical expressions is a fundamental skill in mathematics, and there are several tips and strategies that can make the process more efficient and accurate. Here are some additional techniques to keep in mind:
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Look for Perfect Square/Cube/Higher-Order Factors: The key to simplifying radicals is to identify perfect square factors for square roots, perfect cube factors for cube roots, and so on. This allows you to extract the root of the perfect factor, leaving a simpler expression under the radical.
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Prime Factorization: When dealing with larger radicands, prime factorization is an invaluable tool. Breaking down the radicand into its prime factors helps you identify any perfect square, cube, or higher-order factors that may be present.
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Rationalizing the Denominator: If you have a radical expression in the denominator of a fraction, it's often necessary to rationalize the denominator. This involves multiplying both the numerator and denominator by a suitable expression that eliminates the radical from the denominator.
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Combining Like Radicals: Just like combining like terms in algebra, you can combine like radicals. Like radicals have the same index (the small number indicating the root) and the same radicand. For example, 2√3 + 5√3 = 7√3.
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Using Properties of Radicals: Remember the key properties of radicals, such as:
- √(a ⋅ b) = √a ⋅ √b
- √(a / b) = √a / √b
- √[n]{a^n} = a (if n is odd)
- √[n]{a^n} = |a| (if n is even)
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Practice, Practice, Practice: The more you practice simplifying radical expressions, the more comfortable and proficient you'll become. Work through a variety of problems, and don't be afraid to make mistakes – they're part of the learning process.
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Check Your Answer: After simplifying a radical expression, it's always a good idea to check your answer. You can do this by squaring (for square roots), cubing (for cube roots), or raising to the appropriate power to see if you get back the original radicand.
By incorporating these tips and strategies into your approach, you'll be well-equipped to tackle a wide range of radical simplification problems with confidence.
Conclusion
In conclusion, simplifying radical expressions, particularly those involving cube roots, is a crucial skill in mathematics. The problem we addressed, What is the product of √[3]{24} and √[3]{45}?, provided a practical example of how to apply the principles of radical simplification. Through a step-by-step process, we successfully simplified the expression to 6√[3]{5}, demonstrating the power of mathematical manipulation and the elegance of concise solutions.
We began by establishing a solid understanding of cube roots and radical expressions, laying the foundation for our simplification journey. We then meticulously broke down the expression, combining radicands, factoring, identifying perfect cube factors, and applying the properties of cube roots. This methodical approach allowed us to navigate the complexities of the expression and arrive at the correct answer.
Beyond the specific solution, this exercise highlighted the broader significance of simplifying radical expressions. It showcased how simplification not only leads to more manageable forms but also reveals underlying patterns and relationships within mathematical expressions. The ability to simplify radicals is a valuable asset in various areas of mathematics, empowering us to solve complex problems and gain deeper insights.
Furthermore, we explored additional tips and strategies for simplifying radical expressions, such as looking for perfect factors, utilizing prime factorization, rationalizing denominators, combining like radicals, and leveraging the properties of radicals. These techniques provide a comprehensive toolkit for tackling a wide range of radical simplification challenges.
Ultimately, mastering the art of simplifying radical expressions requires a combination of conceptual understanding, strategic application of techniques, and consistent practice. By embracing these principles, we can unlock the hidden beauty and power of mathematics, paving the way for further exploration and discovery.
