How To Find The Cube Root Of 64x¹²

The realm of mathematics often presents us with intriguing challenges, and one such challenge lies in understanding and simplifying algebraic expressions. Cube roots, in particular, play a significant role in various mathematical contexts, from solving equations to simplifying complex formulas. In this article, we will delve into the process of finding the cube root of the expression 64x¹², providing a step-by-step explanation and exploring the underlying mathematical principles. Our goal is not just to arrive at the correct answer but also to foster a deeper understanding of the concepts involved. We will break down the problem into manageable parts, making it accessible to learners of all levels. By the end of this discussion, you will be equipped with the knowledge and skills to confidently tackle similar problems involving cube roots and algebraic expressions.

Before we dive into the specifics of the problem, let's take a moment to establish a solid foundation by defining what a cube root actually is. In mathematics, the cube root of a number is a value that, when multiplied by itself three times, yields the original number. In other words, if we have a number 'y', its cube root is a number 'x' such that x * x * x = y. This can also be written as x³ = y. The symbol for the cube root is ∛, which is a radical symbol with a small 3 written as a superscript, indicating the cube root. For instance, the cube root of 8 is 2, because 2 * 2 * 2 = 8, or ∛8 = 2. Similarly, the cube root of 27 is 3, as 3 * 3 * 3 = 27. Understanding this fundamental concept is crucial for simplifying expressions involving cube roots.

When dealing with algebraic expressions, we often encounter variables raised to various powers. The same principle of cube roots applies here as well. For example, the cube root of x³ is x, because x * x * x = x³. This understanding extends to more complex expressions involving coefficients and exponents. To find the cube root of an expression like 64x¹², we need to consider both the numerical coefficient (64) and the variable term (x¹²). We will find the cube root of each part separately and then combine the results. This approach allows us to break down the problem into smaller, more manageable steps, making it easier to understand and solve. In the following sections, we will apply this concept to our specific problem and demonstrate how to find the cube root of 64x¹².

To effectively determine the cube root of 64x¹², we need to deconstruct the expression into its fundamental components. This involves examining both the numerical coefficient and the variable term separately. The expression 64x¹² can be seen as a product of two distinct parts: the number 64 and the variable x raised to the power of 12. Our strategy is to find the cube root of each part individually and then combine the results. This approach is based on the property of radicals that states the cube root of a product is equal to the product of the cube roots. Mathematically, this can be expressed as ∛(ab) = ∛a * ∛b. By applying this principle, we can simplify the problem and make it easier to solve. Let's first focus on finding the cube root of the numerical coefficient, 64.

The number 64 is a perfect cube, which means it is the result of cubing an integer. To find the cube root of 64, we need to identify the number that, when multiplied by itself three times, equals 64. Through either recall of common cubes or by prime factorization, we can determine that 4 * 4 * 4 = 64. Therefore, the cube root of 64 is 4, or ∛64 = 4. This is a crucial step in simplifying the original expression. Next, we will turn our attention to the variable term, x¹², and determine its cube root. This involves understanding how exponents behave under radical operations. The cube root of a variable raised to a power can be found by dividing the exponent by 3. This is because the cube root operation is the inverse of cubing, and when we cube a term with an exponent, we multiply the exponent by 3. Therefore, when taking the cube root, we perform the inverse operation, which is division.

Now, let's focus on finding the cube root of 64. As we discussed earlier, the cube root of a number is a value that, when multiplied by itself three times, equals the original number. In this case, we are looking for a number 'x' such that x * x * x = 64, or x³ = 64. There are several ways to approach this. One method is to simply recall common cubes. We know that 1³ = 1, 2³ = 8, 3³ = 27, and 4³ = 64. Thus, we can quickly identify that the cube root of 64 is 4. Another method is to use prime factorization. We can break down 64 into its prime factors: 64 = 2 * 2 * 2 * 2 * 2 * 2 = 2⁶. To find the cube root, we group the factors into sets of three: (2 * 2 * 2) * (2 * 2 * 2) = 2³ * 2³. Taking the cube root of each group, we get 2 * 2 = 4. This confirms that ∛64 = 4.

Understanding the process of finding cube roots of numbers is essential for simplifying more complex expressions. The ability to recognize perfect cubes and to use prime factorization are valuable skills in mathematics. In the context of our problem, finding the cube root of 64 is the first step in simplifying the expression 64x¹². We have successfully determined that ∛64 = 4. Now, we move on to the next part of the expression, which involves finding the cube root of the variable term, x¹². This will require a different approach, as we are dealing with exponents rather than numerical values. However, the same fundamental principles of cube roots apply, and we will use these principles to simplify the variable term. In the following section, we will explore how to find the cube root of x¹² and combine it with the cube root of 64 to arrive at the final answer.

Having successfully found the cube root of the numerical coefficient, 64, which is 4, we now turn our attention to the variable term, x¹². To find the cube root of x¹², we need to understand how exponents behave under radical operations. The rule we apply here is that the cube root of a variable raised to a power is found by dividing the exponent by 3. This is because the cube root operation is the inverse of cubing, and when we cube a term with an exponent, we multiply the exponent by 3. Therefore, when taking the cube root, we perform the inverse operation, which is division. In mathematical terms, ∛(xⁿ) = x^(n/3). Applying this rule to our expression, we have ∛(x¹²) = x^(12/3). Dividing the exponent 12 by 3, we get 4. Therefore, the cube root of x¹² is x⁴.

This result is a direct consequence of the properties of exponents and radicals. It demonstrates how we can simplify algebraic expressions by applying these rules. The ability to manipulate exponents and radicals is a fundamental skill in algebra and is crucial for solving various types of mathematical problems. In the context of our problem, finding the cube root of x¹² is the second key step in simplifying the entire expression 64x¹². We have now successfully found both the cube root of 64 and the cube root of x¹². In the next section, we will combine these results to arrive at the final answer. This will involve multiplying the cube root of the numerical coefficient (4) with the cube root of the variable term (x⁴). This step will complete the process of finding the cube root of 64x¹² and provide us with the simplified expression.

With the individual components now simplified, we can combine our findings to determine the cube root of 64x¹². We previously established that the cube root of 64 is 4 (∛64 = 4) and the cube root of x¹² is x⁴ (∛x¹² = x⁴). To find the cube root of the entire expression, we multiply these individual cube roots together. This is based on the property of radicals that states the cube root of a product is equal to the product of the cube roots, which we mentioned earlier: ∛(ab) = ∛a * ∛b. Applying this principle, we have ∛(64x¹²) = ∛64 * ∛x¹². Substituting the values we found, we get ∛(64x¹²) = 4 * x⁴. Therefore, the cube root of 64x¹² is 4x⁴.

This result represents the simplified form of the original expression. It demonstrates how we can break down a complex problem into smaller, more manageable parts and then combine the results to arrive at the solution. The ability to simplify expressions involving radicals and exponents is a crucial skill in algebra and is essential for solving various mathematical problems. In this case, we have successfully found the cube root of 64x¹² by first finding the cube root of the numerical coefficient and the variable term separately, and then multiplying the results together. This approach is applicable to a wide range of similar problems and provides a systematic way to simplify expressions involving cube roots. In the final section, we will summarize our findings and provide a concise answer to the original question.

In summary, we embarked on a journey to find the cube root of the expression 64x¹². We began by defining what a cube root is and establishing the fundamental principles of how cube roots interact with numerical coefficients and variable terms. We then deconstructed the expression into its individual components: the numerical coefficient 64 and the variable term x¹². We found the cube root of 64 to be 4 and the cube root of x¹² to be x⁴. Finally, we combined these results by multiplying them together, arriving at the solution: 4x⁴. Therefore, the cube root of 64x¹² is 4x⁴.

This process highlights the importance of understanding the properties of radicals and exponents in simplifying algebraic expressions. By breaking down a complex problem into smaller, more manageable steps, we can effectively solve it. The ability to manipulate algebraic expressions and to find cube roots is a fundamental skill in mathematics and is essential for various applications in science, engineering, and other fields. We hope that this detailed explanation has provided a clear understanding of how to find the cube root of 64x¹² and has equipped you with the knowledge and skills to tackle similar problems with confidence. The correct answer among the options provided is A. 4x⁴.