Solving The Cube Root Equation $\sqrt[3]{x-1}=-4$

At its core, mathematics thrives on problem-solving, and equations form the backbone of this pursuit. This article delves into a particular equation involving a cube root and guides you through the process of finding its solution. More specifically, we'll dissect the equation $\sqrt[3]{x-1} = -4$. Our primary objective is to determine the value of 'x' that satisfies this equation. We will systematically explore the steps required to isolate 'x' and arrive at the correct answer. This exploration isn't just about finding the answer; it's about understanding the underlying principles of algebraic manipulation and equation-solving. We aim to provide a clear and concise explanation that will empower you to tackle similar problems with confidence. The question presented offers a set of potential solutions, and we will rigorously test each one against the original equation. This methodical approach is crucial in mathematics, where precision and accuracy are paramount. By the end of this article, you will have a solid grasp of how to solve cube root equations and will be well-equipped to handle more complex mathematical challenges. So, let's embark on this journey of mathematical discovery and unlock the solution to this intriguing equation. Remember, mathematics is not just about formulas and rules; it's about logical thinking and problem-solving strategies. As we delve deeper into the problem, we will emphasize these aspects, fostering a more profound understanding of the subject.

Understanding Cube Roots

Before we directly jump into the process of solving the equation $\sqrt[3]{x-1} = -4$, it is crucial to have a solid grasp of what cube roots entail. The 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 * 2 * 2 = 8. Similarly, the cube root of -27 is -3 because (-3) * (-3) * (-3) = -27. Unlike square roots, which only deal with non-negative numbers in the realm of real numbers, cube roots can handle both positive and negative numbers. This is because a negative number multiplied by itself three times results in a negative number. Understanding this fundamental concept is essential for solving equations involving cube roots. It allows us to confidently work with negative values under the cube root symbol without any ambiguity. The cube root operation is the inverse of cubing a number. Just as squaring a number and then taking its square root returns the original number, cubing a number and then taking its cube root also returns the original number. This inverse relationship is crucial in solving equations where we need to isolate a variable. In our equation, the cube root operation is acting on the expression (x - 1). To isolate 'x', we will need to reverse this operation by cubing both sides of the equation. This understanding of cube roots forms the foundation for the subsequent steps in our problem-solving journey. Without a clear understanding of this concept, the rest of the solution process may seem confusing or arbitrary. Therefore, before moving on, make sure you are comfortable with the idea of cube roots and their properties. Remember, mathematics is a building block subject, and each concept builds upon the previous ones. A strong foundation in cube roots will greatly enhance your ability to solve more advanced equations in the future.

Isolating the Variable

The primary goal in solving any equation is to isolate the variable, in this case, 'x'. This means manipulating the equation until 'x' stands alone on one side, giving us its value. In our equation, $\sqrt[3]{x-1} = -4$, the variable 'x' is trapped inside a cube root. To free 'x', we need to eliminate the cube root. The most direct way to do this is to perform the inverse operation, which is cubing. We cube both sides of the equation to maintain the balance. This is a fundamental principle in algebra: whatever operation you perform on one side of the equation, you must also perform on the other side to preserve equality. Cubing both sides of the equation gives us $(\sqrt[3]{x-1})^3 = (-4)^3$. On the left side, the cube and the cube root cancel each other out, leaving us with x - 1. On the right side, we have (-4)^3, which means -4 multiplied by itself three times: (-4) * (-4) * (-4) = -64. So, our equation now simplifies to x - 1 = -64. We are one step closer to isolating 'x'. Now, we need to get rid of the -1 on the left side. We do this by adding 1 to both sides of the equation. This gives us x - 1 + 1 = -64 + 1. The -1 and +1 on the left side cancel each other out, leaving us with just 'x'. On the right side, -64 + 1 equals -63. Therefore, we arrive at our solution: x = -63. This methodical process of isolating the variable is a core skill in algebra. It requires a clear understanding of inverse operations and the principle of maintaining equality. By carefully applying these principles, we have successfully solved for 'x' in our equation. This step-by-step approach not only leads us to the correct answer but also builds a strong foundation for tackling more complex equations in the future.

Verifying the Solution

Once we've arrived at a potential solution, it's crucial to verify its correctness. This step is often overlooked but is essential to ensure accuracy and catch any potential errors made during the solving process. To verify our solution, x = -63, we substitute this value back into the original equation: $\sqrt[3]{x-1} = -4$. Replacing 'x' with -63, we get $\sqrt[3]{-63-1} = -4$. Simplifying the expression inside the cube root, we have $\sqrt[3]{-64} = -4$. Now, we need to determine the cube root of -64. As we discussed earlier, the cube root of a negative number is a negative number. The cube root of -64 is -4 because (-4) * (-4) * (-4) = -64. Therefore, the left side of the equation simplifies to -4. Comparing this to the right side of the equation, which is also -4, we see that both sides are equal. This confirms that our solution, x = -63, is indeed correct. Verifying the solution is not just about confirming the answer; it's also about reinforcing our understanding of the problem and the steps we took to solve it. It provides a sense of closure and confidence in our mathematical abilities. In this case, by substituting our solution back into the original equation and verifying that it holds true, we have strengthened our understanding of cube root equations and the process of solving them. This practice of verification is a valuable habit to cultivate in mathematics. It not only ensures accuracy but also deepens our understanding of the subject matter. So, always remember to verify your solutions whenever possible, especially in more complex problems where errors are more likely to occur.

The Correct Answer

After systematically isolating the variable 'x' in the equation $\sqrt[3]{x-1} = -4$ and subsequently verifying our solution, we have arrived at the definitive answer. Our calculations led us to the solution x = -63. We then meticulously verified this solution by substituting it back into the original equation and confirming that it holds true. This rigorous process ensures that we have not only found a solution but also that the solution is accurate and satisfies the given equation. Looking back at the original question, which presented a set of potential answers, we can now confidently identify the correct one. The options provided were:

• A. -11
• B. -63
• C. -65
• D. no solution

Based on our calculations and verification, it is clear that the correct answer is B. -63. The other options are incorrect because they do not satisfy the original equation when substituted for 'x'. Option A, -11, would result in a cube root of a negative number that is not -4. Option C, -65, would also result in a cube root of a negative number that is not -4. Option D, "no solution," is incorrect because we have successfully found a solution and verified its correctness. This process of elimination further reinforces the validity of our solution. By carefully considering each option and comparing it to our calculated solution, we can be absolutely certain that we have selected the correct answer. This methodical approach is crucial in mathematics, where precision and accuracy are paramount. Therefore, the final and correct answer to the equation $\sqrt[3]{x-1} = -4$ is x = -63, corresponding to option B. This concludes our journey of solving this cube root equation, demonstrating the power of algebraic manipulation and the importance of verification.

In this comprehensive guide, we have successfully navigated the process of solving the equation $\sqrt[3]{x-1} = -4$. We began by establishing a solid understanding of cube roots, emphasizing their ability to handle both positive and negative numbers, a key distinction from square roots. This foundational knowledge paved the way for effectively tackling the equation at hand. We then moved on to the crucial step of isolating the variable 'x'. This involved applying the inverse operation of cubing to both sides of the equation, a technique rooted in the fundamental principle of maintaining equality in algebraic manipulations. By carefully cubing both sides, we eliminated the cube root, bringing us closer to our goal of isolating 'x'. We meticulously simplified the equation, step by step, ensuring that each operation was performed accurately. This methodical approach is a hallmark of effective problem-solving in mathematics. After isolating 'x', we arrived at the potential solution x = -63. However, we did not stop there. We recognized the paramount importance of verification in mathematics and diligently substituted our solution back into the original equation. This crucial step allowed us to confirm the correctness of our answer and catch any potential errors that might have occurred during the solving process. The verification process involved evaluating the cube root of -64, which we correctly identified as -4, thus confirming that our solution indeed satisfied the equation. Finally, we reviewed the given options and confidently selected the correct answer, B. -63. This process of elimination further reinforced our confidence in the accuracy of our solution. Throughout this journey, we have not only solved a specific equation but also reinforced essential mathematical principles and problem-solving strategies. We have emphasized the importance of understanding fundamental concepts, applying inverse operations, maintaining equality, and verifying solutions. These skills are transferable to a wide range of mathematical problems and will serve you well in your future mathematical endeavors. Remember, mathematics is not just about finding answers; it's about developing logical thinking and problem-solving skills. By embracing a methodical and rigorous approach, you can unlock the power of mathematics and confidently tackle any challenge that comes your way.