Cube Root Of 27x¹⁸ A Step-by-Step Solution

In the realm of mathematics, understanding the concept of cube roots is fundamental. This exploration delves into the process of finding the cube root of the expression 27x¹⁸, providing a detailed explanation and step-by-step solution. We will dissect the components of the expression, apply the properties of exponents and radicals, and arrive at the correct answer, ensuring a clear understanding of the underlying principles. The exploration of cube roots extends beyond mere calculations; it involves grasping the relationship between numbers and their roots, a concept that is crucial in various mathematical and scientific applications. Before we jump into solving this specific problem, let's first understand what a cube root is. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. Similarly, the cube root of 27 is 3 because 3 * 3 * 3 = 27. Understanding this basic concept is crucial for tackling more complex expressions involving cube roots and variables. Moreover, recognizing perfect cubes is a helpful skill in simplifying cube root expressions. Perfect cubes are numbers that can be obtained by cubing an integer. Examples of perfect cubes include 1 (1³), 8 (2³), 27 (3³), 64 (4³), and so on. Identifying these numbers within a larger expression can significantly simplify the process of finding the cube root. The properties of exponents also play a critical role in simplifying expressions involving cube roots, especially when dealing with variables raised to powers. When taking the cube root of a variable raised to a power, we divide the exponent by 3. For instance, the cube root of x⁶ is x², because 6 divided by 3 equals 2. This rule is a direct consequence of the laws of exponents and is essential for simplifying expressions like 27x¹⁸.

The expression 27x¹⁸ comprises two primary components: the numerical coefficient 27 and the variable term x¹⁸. To find the cube root of this expression, we must address each component separately, leveraging our understanding of cube roots and exponent rules. The first component, 27, is a perfect cube. As we established earlier, the cube root of 27 is 3, since 3 multiplied by itself three times (3 * 3 * 3) equals 27. Recognizing this immediately simplifies the problem. Now, let's turn our attention to the variable term, x¹⁸. This term consists of the variable 'x' raised to the power of 18. To find the cube root of x¹⁸, we need to apply the rule that states when taking the cube root of a variable raised to a power, we divide the exponent by 3. In this case, we divide 18 by 3, which yields 6. Therefore, the cube root of x¹⁸ is x⁶. This step is crucial in simplifying the expression and is a direct application of the properties of exponents and radicals. The ability to break down complex expressions into their constituent parts is a fundamental skill in mathematics. By isolating the numerical coefficient and the variable term, we can apply the appropriate rules and techniques to each part, making the problem more manageable. Furthermore, understanding the structure of the expression allows us to see the underlying mathematical relationships more clearly. This approach is not only helpful in solving this specific problem but also in tackling a wide range of mathematical challenges. Remember, the key is to identify the individual components and then apply the relevant mathematical principles to each one.

Now, let's consolidate our findings and present a step-by-step solution to determine the cube root of 27x¹⁸. The process involves applying the cube root to both the numerical coefficient and the variable term individually. Step 1: Identify the cube root of the numerical coefficient. We know that the cube root of 27 is 3, as 3 * 3 * 3 = 27. This is a straightforward application of our knowledge of perfect cubes. Step 2: Determine the cube root of the variable term. To find the cube root of x¹⁸, we divide the exponent (18) by 3, which gives us 6. Therefore, the cube root of x¹⁸ is x⁶. This step utilizes the property of exponents that allows us to simplify radical expressions. Step 3: Combine the results. We now have the cube root of the numerical coefficient (3) and the cube root of the variable term (x⁶). To find the cube root of the entire expression, we simply multiply these two results together. This gives us 3 * x⁶, which is commonly written as 3x⁶. Thus, the cube root of 27x¹⁸ is 3x⁶. This step-by-step approach not only provides the solution but also illustrates the logical progression involved in solving the problem. By breaking down the problem into smaller, manageable steps, we can avoid confusion and ensure accuracy. Moreover, this method can be applied to a variety of similar problems involving cube roots and exponents. Remember, the key to success in mathematics lies in understanding the fundamental principles and applying them systematically.

To ensure the accuracy of our solution, it's crucial to verify that the cube of 3x⁶ indeed equals 27x¹⁸. This process involves cubing the expression 3x⁶ and comparing the result with the original expression. To cube 3x⁶, we multiply it by itself three times: (3x⁶) * (3x⁶) * (3x⁶). Let's break this down step by step. First, we multiply the numerical coefficients: 3 * 3 * 3 = 27. Next, we multiply the variable terms. When multiplying variables with exponents, we add the exponents: x⁶ * x⁶ * x⁶ = x^(6+6+6) = x¹⁸. Now, we combine the numerical coefficient and the variable term: 27 * x¹⁸ = 27x¹⁸. As we can see, the cube of 3x⁶ is indeed 27x¹⁸, which confirms that our solution is correct. Verification is an essential step in problem-solving, especially in mathematics. It allows us to catch any potential errors and ensures that we have arrived at the correct answer. This process not only provides confidence in our solution but also reinforces our understanding of the underlying mathematical principles. Furthermore, the method of verification can vary depending on the type of problem. In this case, we used the definition of a cube root to verify our solution. However, in other scenarios, we might use different techniques, such as substituting values or using inverse operations.

Based on our step-by-step solution, we have determined that the cube root of 27x¹⁸ is 3x⁶. Now, let's examine the provided answer choices to identify the correct option. The answer choices are:

A. 3x⁶ B. 3x⁹ C. 9x⁶ D. 9x⁹

Comparing our solution (3x⁶) with the answer choices, we can clearly see that option A, 3x⁶, matches our result. Therefore, option A is the correct answer. The other options are incorrect due to errors in either the numerical coefficient or the exponent of the variable term. For instance, option B, 3x⁹, has the correct numerical coefficient but an incorrect exponent. Option C, 9x⁶, has the correct exponent but an incorrect numerical coefficient. Option D, 9x⁹, has errors in both the numerical coefficient and the exponent. This comparison highlights the importance of careful calculation and attention to detail when solving mathematical problems. Even a small error can lead to an incorrect answer. Therefore, it's crucial to double-check each step and ensure that all calculations are accurate. Moreover, understanding the underlying concepts and principles can help us identify and avoid common mistakes. In this case, a solid understanding of cube roots and exponent rules is essential for arriving at the correct solution and recognizing the incorrect options.

In conclusion, the cube root of 27x¹⁸ is 3x⁶. This problem serves as a valuable exercise in understanding and applying the concepts of cube roots, exponents, and algebraic manipulation. By breaking down the expression into its components, applying the appropriate rules, and verifying the solution, we have demonstrated a systematic approach to solving this type of problem. The ability to find cube roots of expressions involving variables and exponents is a fundamental skill in algebra and is essential for tackling more advanced mathematical concepts. Moreover, the problem-solving techniques used in this exploration can be applied to a wide range of mathematical challenges. The key is to understand the underlying principles, break down complex problems into smaller steps, and verify the solution to ensure accuracy. Mastery of these skills not only leads to success in mathematics but also develops critical thinking and problem-solving abilities that are valuable in various aspects of life. Remember, mathematics is not just about memorizing formulas and procedures; it's about understanding the relationships between numbers and concepts and applying that knowledge to solve problems. This exploration of the cube root of 27x¹⁸ has hopefully provided a clear and comprehensive understanding of the process involved and has inspired a deeper appreciation for the beauty and power of mathematics.