Solving $-2 \sqrt[3]{C}+19=15$ A Step-by-Step Guide

\sqrt[3]{C}+19=15$

In the realm of mathematics, particularly algebra, a fundamental skill is the ability to manipulate equations to solve for a specific variable. This involves isolating the variable on one side of the equation, effectively expressing its value in terms of the other constants and variables present. The equation presented, $-2 \sqrt[3]{C}+19=15$, provides an excellent example of this process, where our goal is to determine the value of 'C'. This article will provide a comprehensive, step-by-step guide to solving this equation, ensuring clarity and understanding at each stage. We will delve into the underlying principles of algebraic manipulation, highlighting the importance of maintaining equality while performing operations on both sides of the equation. By the end of this guide, you will not only be able to solve this specific equation but also gain a broader understanding of how to approach similar problems in algebra. The techniques discussed here form the bedrock of more advanced mathematical concepts, making this a crucial skill for anyone pursuing further studies in mathematics, science, or engineering. Our approach will be methodical, starting with the initial isolation of the term containing 'C' and progressing through the removal of the cube root and the final calculation of 'C'. This systematic method will help solidify your understanding of the process and allow you to apply these techniques to a wide range of algebraic equations. Furthermore, we will emphasize the importance of checking your solution to ensure accuracy, a practice that is essential in mathematical problem-solving. The ultimate aim is to empower you with the knowledge and skills to confidently tackle algebraic equations and understand the logical steps involved in arriving at the correct solution. Let's embark on this journey of mathematical discovery and unravel the value of 'C' in this intriguing equation.

Step 1: Isolate the Cube Root Term

To begin solving for 'C' in the equation $-2 \sqrt[3]{C}+19=15$, our initial objective is to isolate the term that contains 'C', which is $-2 \sqrt[3]{C}$. This is achieved by systematically removing any terms that are added or subtracted from this expression on the left-hand side of the equation. In this case, we have '+19' being added to the cube root term. To counteract this, we apply the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation to maintain equality. Therefore, we subtract 19 from both sides of the equation. This crucial step ensures that the equation remains balanced, preserving the relationship between the two sides. The equation now transforms as follows:

$-2 \sqrt[3]{C}+19 - 19 = 15 - 19$

This simplifies to:

$-2 \sqrt[3]{C} = -4$

By subtracting 19 from both sides, we have successfully isolated the term containing the cube root of 'C'. This is a significant milestone in our problem-solving journey, as it brings us closer to isolating 'C' itself. The next step will involve dealing with the coefficient of the cube root term, which is -2. This will require another application of the principle of maintaining equality while performing operations on both sides. The importance of this initial isolation step cannot be overstated; it sets the stage for the subsequent steps and allows us to focus our efforts on the core challenge of finding 'C'. As we proceed, we will continue to emphasize the logical progression of steps and the underlying algebraic principles that guide our actions. This approach not only helps in solving the current equation but also builds a strong foundation for tackling more complex algebraic problems in the future.

Step 2: Divide by the Coefficient

Having successfully isolated the term $-2 \sqrt[3]{C}$ in the previous step, our next task is to eliminate the coefficient -2 that is multiplying the cube root. To achieve this, we again rely on the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation. In this instance, since -2 is multiplying the cube root, we will divide both sides of the equation by -2. This operation will effectively cancel out the -2 on the left-hand side, bringing us closer to isolating the cube root of 'C'. The equation we have from the previous step is:

$-2 \sqrt[3]{C} = -4$

Dividing both sides by -2, we get:

$\frac{-2 \sqrt[3]{C}}{-2} = \frac{-4}{-2}$

This simplifies to:

$\sqrt[3]{C} = 2$

By dividing both sides by -2, we have successfully isolated the cube root of 'C'. This is a crucial step because it allows us to now focus solely on eliminating the cube root operation to find the value of 'C'. The division operation here highlights the importance of understanding inverse operations in algebra. Multiplication and division are inverse operations, meaning that dividing by a number undoes the effect of multiplying by that number. This principle is fundamental to solving equations and is used extensively in various mathematical contexts. With the cube root of 'C' now isolated, we are well-positioned to move on to the next step, which will involve using the inverse operation of the cube root to finally solve for 'C'. The methodical approach we are taking, step by step, ensures that each operation is performed with clarity and precision, minimizing the chances of error and maximizing our understanding of the process.

Step 3: Cube Both Sides of the Equation

Now that we have isolated the cube root of 'C', represented as $\sqrt[3]{C} = 2$, the next crucial step is to eliminate the cube root itself. To do this, we employ the inverse operation of taking the cube root, which is cubing. Just as squaring a square root cancels out the radical, cubing a cube root will effectively remove the radical symbol, leaving us with the variable 'C' on its own. Again, we must adhere to the fundamental principle of maintaining equality by performing the same operation on both sides of the equation. This ensures that the balance of the equation is preserved, and the solution we obtain remains valid. Starting with the equation:

$\sqrt[3]{C} = 2$

We cube both sides:

$(\sqrt[3]{C})^3 = 2^3$

The cube root and the cube operation cancel each other out on the left-hand side, leaving us with 'C'. On the right-hand side, we calculate 2 cubed, which is 2 multiplied by itself three times (2 * 2 * 2), resulting in 8. Therefore, the equation simplifies to:

$C = 8$

By cubing both sides, we have successfully eliminated the cube root and solved for 'C'. This step demonstrates the power of using inverse operations to isolate variables in equations. The ability to recognize and apply inverse operations is a cornerstone of algebraic problem-solving. With 'C' now isolated, we have arrived at a potential solution, but it is always prudent to verify our answer to ensure accuracy. The next step will involve substituting our calculated value of 'C' back into the original equation to confirm that it satisfies the equation. This verification process is a critical part of the problem-solving process, helping to catch any potential errors and build confidence in our solution.

Step 4: Verify the Solution

Having solved for 'C' and obtained the value of 8, it is crucial to verify our solution. This step is not merely a formality; it is a vital part of the problem-solving process that ensures the accuracy of our answer. Verification involves substituting the calculated value of 'C' back into the original equation and checking if both sides of the equation are equal. If the equation holds true, then our solution is correct. If not, it indicates that an error was made during the solving process, and we need to revisit our steps to identify and correct the mistake. The original equation we started with is:

$-2 \sqrt[3]{C}+19=15$

Now, we substitute C = 8 into the equation:

$-2 \sqrt[3]{8}+19=15$

First, we evaluate the cube root of 8. The cube root of 8 is 2, because 2 * 2 * 2 = 8. So, the equation becomes:

$-2 * 2 + 19 = 15$

Next, we perform the multiplication:

$-4 + 19 = 15$

Finally, we perform the addition:

$15 = 15$

Since both sides of the equation are equal, our solution C = 8 is verified and confirmed to be correct. This verification process underscores the importance of meticulousness in mathematical problem-solving. By taking the time to check our answer, we can be confident in the accuracy of our solution and avoid potential errors. The verification step also reinforces our understanding of the equation and the relationships between the variables and constants involved. In this case, the successful verification of C = 8 provides closure to the problem-solving process and solidifies our understanding of how to solve equations involving cube roots.

Conclusion

In conclusion, we have successfully solved the equation $-2 \sqrt[3]{C}+19=15$ for the variable 'C'. Through a systematic, step-by-step approach, we have demonstrated the key principles of algebraic manipulation, including isolating terms, using inverse operations, and maintaining equality throughout the process. The solution we arrived at is C = 8, which we rigorously verified by substituting it back into the original equation. This journey through the solution process highlights the importance of a methodical approach to problem-solving in mathematics. Each step, from isolating the cube root term to cubing both sides of the equation, was performed with a clear understanding of the underlying algebraic principles. The emphasis on maintaining equality by performing the same operation on both sides of the equation was paramount, ensuring that our solution remained valid at every stage. Furthermore, the verification step underscored the significance of checking our work to ensure accuracy and build confidence in our answer. The techniques and principles demonstrated in this guide are not limited to this specific equation; they are widely applicable to a range of algebraic problems. The ability to manipulate equations, isolate variables, and use inverse operations is a fundamental skill in mathematics and is essential for further studies in science, engineering, and other related fields. By mastering these techniques, you can confidently tackle a wide variety of mathematical challenges and develop a deeper understanding of the relationships between variables and constants. The satisfaction of arriving at a correct solution, coupled with the knowledge gained along the way, makes the study of mathematics a rewarding and enriching experience. We encourage you to continue practicing these techniques and exploring new mathematical concepts to further enhance your problem-solving abilities.