Solving The Equation 7 - Cube Root Of 2-x = 12 A Step-by-Step Guide
In the realm of mathematics, equations serve as fundamental tools for expressing relationships between variables and constants. Among the diverse types of equations, those involving radicals, particularly cube roots, often pose a unique challenge to students. This article delves into the intricacies of solving equations containing radicals, providing a step-by-step guide to unravel the complexities and arrive at accurate solutions. We will dissect the equation $7 - \sqrt[3]{2 - x} = 12$, illustrating the process of isolating the radical, eliminating it through exponentiation, and ultimately determining the value of the unknown variable, x. Mastering the techniques presented here will empower you to confidently tackle a wide range of radical equations, enhancing your problem-solving prowess in mathematics. This exploration will not only focus on the procedural steps but also on the underlying mathematical principles that govern the manipulation of equations and radicals. Understanding these concepts is crucial for developing a robust mathematical foundation and for applying these skills to more advanced mathematical problems. Throughout this article, we will emphasize the importance of accuracy and attention to detail, as these are paramount in ensuring the correctness of your solutions. By the end of this guide, you will be well-equipped to approach radical equations with confidence and precision.
Understanding Radical Equations
Radical equations are mathematical expressions where the variable is found inside a radical, such as a square root, cube root, or any higher-order root. The presence of the radical necessitates a specific approach to solving the equation, one that involves isolating the radical term and then eliminating it by raising both sides of the equation to a power that corresponds to the index of the radical. This process effectively "undoes" the radical, allowing us to work with a more conventional algebraic expression. However, it's crucial to note that this step can sometimes introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. Therefore, it's always necessary to check the solutions obtained against the original equation to ensure their validity. In the context of the equation $7 - \sqrt[3]{2 - x} = 12$, the cube root term presents a unique characteristic compared to square roots. Cube roots can accept any real number as input, whether positive or negative, whereas square roots are typically defined only for non-negative numbers in the real number system. This distinction influences the potential solutions and the steps involved in verifying them. Our approach will involve careful algebraic manipulation, paying close attention to maintaining the equality of both sides of the equation throughout the process. We will also highlight the importance of understanding the domain of the radical function, which is the set of all possible input values for which the function is defined. This understanding is crucial for identifying potential extraneous solutions and ensuring the accuracy of our final answer.
Step-by-Step Solution to $7 - \sqrt[3]{2 - x} = 12$
To effectively solve the equation $7 - \sqrt[3]{2 - x} = 12$, we embark on a meticulous, step-by-step process that begins with isolating the radical term. This involves strategically maneuvering the equation to bring the cube root expression to one side while keeping all other terms on the opposite side. By subtracting 7 from both sides, we initiate this isolation, resulting in the equation $\sqrt[3]{2 - x} = -5$. This step is crucial as it sets the stage for eliminating the cube root. The next pivotal step is to eliminate the cube root itself. To achieve this, we employ the inverse operation of taking a cube root, which is cubing. Cubing both sides of the equation ensures that we maintain the balance of the equation while simultaneously removing the radical. This transformation yields $(-\sqrt[3]{2 - x})^3 = (-5)^3$, which simplifies to $2 - x = -125$. Now, we have a linear equation that is much simpler to solve. Our focus shifts to isolating the variable x. By subtracting 2 from both sides, we further isolate the term containing x, leading to the equation $-x = -127$. Finally, to solve for x, we multiply both sides by -1, revealing the solution $x = 127$. However, our journey doesn't conclude here. To ensure the validity of our solution, it's imperative to check it against the original equation. This step is particularly crucial when dealing with radical equations, as the process of raising both sides to a power can sometimes introduce extraneous solutions. By substituting $x = 127$ back into the original equation, we can verify its correctness. This careful verification process is a cornerstone of accurate problem-solving in mathematics, safeguarding against errors and ensuring the reliability of our results.
1. Isolate the Radical
Our primary objective in tackling this equation is to isolate the radical term, which in this case is the cube root expression, $\sqrt[3]{2 - x}$. This isolation is a critical first step because it allows us to effectively eliminate the radical in the subsequent steps. To achieve this, we strategically manipulate the equation to bring the radical term to one side while moving all other terms to the opposite side. In the given equation, $7 - \sqrt[3]{2 - x} = 12$, we begin by subtracting 7 from both sides. This operation maintains the equality of the equation while simultaneously separating the radical term. Subtracting 7 from both sides yields $7 - \sqrt[3]{2 - x} - 7 = 12 - 7$, which simplifies to $\sqrt[3]{2 - x} = -5$. Now, the cube root term is isolated on the left side of the equation, setting the stage for the next phase of our solution. This step exemplifies the importance of algebraic manipulation in equation solving, where strategic operations are employed to simplify the equation and bring it closer to a solvable form. The isolation of the radical term is not merely a procedural step; it reflects a deeper understanding of the structure of the equation and the relationships between its components. By isolating the radical, we are essentially preparing it for elimination, which is the key to unlocking the value of the unknown variable, x. This methodical approach is characteristic of effective problem-solving in mathematics, where complex problems are broken down into smaller, more manageable steps.
2. Eliminate the Radical
With the radical term now isolated, our next decisive move is to eliminate it from the equation. This is a pivotal step that allows us to transition from an equation containing a radical to a more straightforward algebraic form. To eliminate the cube root, we employ the inverse operation, which is cubing. Cubing both sides of the equation ensures that we maintain the balance and equality of the expression while simultaneously undoing the effect of the cube root. Starting with the equation $\sqrt[3]{2 - x} = -5$, we cube both sides, resulting in $(-\sqrt[3]{2 - x})^3 = (-5)^3$. This operation effectively removes the cube root on the left side, leaving us with the expression $2 - x$ on that side. On the right side, we have $(-5)^3$, which evaluates to -125. Therefore, the equation transforms into $2 - x = -125$. This transformation is a significant milestone in our solution process. By eliminating the radical, we have simplified the equation to a linear form, which is far easier to solve. The act of cubing both sides is not merely a mechanical step; it demonstrates a fundamental understanding of the relationship between radicals and exponents. It highlights the principle that raising a radical to a power that matches its index will effectively cancel out the radical, leaving the radicand (the expression inside the radical) exposed. This principle is crucial for solving a wide range of radical equations, and mastering it is essential for success in algebra and beyond.
3. Solve for x
Having successfully eliminated the radical, we now find ourselves with a linear equation, a familiar and easily solvable form. The equation we've arrived at is $2 - x = -125$, and our objective is to isolate the variable x to determine its value. To achieve this, we employ a series of algebraic manipulations, carefully maintaining the balance of the equation at each step. Our first step in isolating x is to subtract 2 from both sides of the equation. This operation effectively moves the constant term from the left side to the right side, bringing us closer to isolating the variable. Subtracting 2 from both sides yields $2 - x - 2 = -125 - 2$, which simplifies to $-x = -127$. Now, the equation is in a much simpler form, with only the term containing x on the left side. However, we still have a negative sign associated with x, which we need to address. To remove the negative sign, we multiply both sides of the equation by -1. This operation changes the sign of both terms, effectively solving for x. Multiplying both sides by -1 gives us $(-1)(-x) = (-1)(-127)$, which simplifies to $x = 127$. We have now arrived at a potential solution for x, which is 127. However, our journey is not yet complete. It's crucial to remember that when solving radical equations, it's imperative to check our solution against the original equation to ensure its validity. This step is particularly important because the process of raising both sides of an equation to a power can sometimes introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one.
4. Check the Solution
The final and arguably most critical step in solving radical equations is to meticulously check the solution we've obtained. This step is not merely a formality; it's an essential safeguard against extraneous solutions, which can arise when we raise both sides of an equation to a power. Extraneous solutions are values that satisfy the transformed equation but do not satisfy the original equation. To check our solution, $x = 127$, we substitute it back into the original equation, $7 - \sqrt[3]2 - x} = 12$. Substituting $x = 127$ into the equation, we get $7 - \sqrt[3]{2 - 127} = 12$. Now, we simplify the expression step by step. First, we evaluate the expression inside the cube root = 12$. Next, we evaluate the cube root of -125, which is -5. This gives us $7 - (-5) = 12$. Simplifying further, we have $7 + 5 = 12$, which simplifies to $12 = 12$. This final equality confirms that our solution, $x = 127$, is indeed valid and satisfies the original equation. The process of checking the solution is a testament to the rigor and precision required in mathematics. It underscores the importance of not only finding a potential solution but also verifying its correctness. This step ensures that our answer is not only mathematically sound but also logically consistent with the original problem. By diligently checking our solutions, we cultivate a deeper understanding of the problem-solving process and develop a habit of accuracy that will serve us well in all our mathematical endeavors.
Conclusion
In conclusion, solving radical equations requires a systematic approach, careful algebraic manipulation, and a thorough understanding of the underlying mathematical principles. In this article, we have meticulously dissected the process of solving the equation $7 - \sqrt[3]{2 - x} = 12$, providing a step-by-step guide that encompasses isolating the radical, eliminating it through exponentiation, solving for the variable, and crucially, checking the solution for validity. The solution we arrived at, $x = 127$, was rigorously verified, affirming its correctness and highlighting the importance of this final step in the problem-solving process. The techniques and strategies discussed here are not limited to this specific equation; they are broadly applicable to a wide range of radical equations. By mastering these techniques, you can confidently tackle complex mathematical problems and enhance your problem-solving abilities. The journey of solving radical equations is not just about finding the answer; it's about developing a deeper understanding of mathematical concepts and cultivating a methodical approach to problem-solving. The skills you acquire in this process will extend far beyond the realm of radical equations, empowering you to excel in various areas of mathematics and beyond. Remember, accuracy, attention to detail, and a commitment to verifying your solutions are the hallmarks of a successful problem solver. As you continue your mathematical journey, embrace these principles, and you will undoubtedly achieve mastery in this fascinating field.
