Solving -11√[5]{D} - 11 = 44 For D A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of solving the equation -11√[5]{D} - 11 = 44 for the variable D. This equation involves a fifth root, which may seem intimidating at first, but with a systematic approach, we can isolate D and find its value. We will break down each step, providing clear explanations and mathematical reasoning to ensure a comprehensive understanding. Our goal is not only to solve this specific equation but also to equip you with the skills and knowledge to tackle similar algebraic challenges. We will emphasize the importance of order of operations, inverse operations, and careful manipulation of equations to arrive at the correct solution. Additionally, we will address the rounding of answers to two decimal places, as required, ensuring that our final result adheres to the specified precision. By the end of this article, you will be confident in your ability to solve equations involving radicals and understand the underlying principles that govern algebraic problem-solving. This equation, like many in algebra, requires a careful step-by-step approach to isolate the variable. Understanding the order of operations and how to apply inverse operations is crucial. Let’s begin by outlining the steps we will take to solve this equation and then dive into the detailed solution. This methodical approach is key to avoiding common errors and ensuring accuracy in our final answer.

Step-by-Step Solution

Step 1: Isolate the Radical Term

The first crucial step in solving any equation involving radicals is to isolate the radical term. In our equation, -11√[5]{D} - 11 = 44, the radical term is -11√[5]{D}. To isolate this term, we need to eliminate the -11 that is added to it. We can achieve this by adding 11 to both sides of the equation. This maintains the balance of the equation and allows us to move closer to isolating the radical. Adding 11 to both sides is an application of the addition property of equality, a fundamental principle in algebra. This step is critical because it simplifies the equation and sets the stage for the next operation. The isolation of the radical term is a common strategy in solving equations with roots, and mastering this step will be beneficial in tackling more complex problems. It is also important to note that performing the same operation on both sides of the equation ensures that the equality remains valid.

  • Original Equation: -11√[5]{D} - 11 = 44
  • Add 11 to both sides: -11√[5]{D} - 11 + 11 = 44 + 11
  • Simplified: -11√[5]{D} = 55

Step 2: Divide by the Coefficient

After isolating the radical term, our next step involves dealing with the coefficient that multiplies the radical. In our equation, -11√[5]{D} = 55, the coefficient is -11. To further isolate the fifth root, we need to divide both sides of the equation by this coefficient. This operation utilizes the division property of equality, another key principle in algebraic manipulations. Dividing both sides by -11 will cancel out the -11 on the left side, leaving us with the fifth root term by itself. This step is crucial because it simplifies the equation even further, bringing us closer to isolating the variable D. By dividing both sides by the coefficient, we effectively undo the multiplication that is currently affecting the radical term. This methodical approach is essential for solving equations involving radicals and ensures that each step contributes to the overall simplification of the problem.

  • Current Equation: -11√[5]{D} = 55
  • Divide both sides by -11: (-11√[5]{D}) / -11 = 55 / -11
  • Simplified: √[5]{D} = -5

Step 3: Eliminate the Fifth Root

Now that we have isolated the fifth root in the equation √[5]{D} = -5, our next task is to eliminate the radical. To do this, we need to raise both sides of the equation to the power that matches the index of the root. In this case, the index is 5, so we will raise both sides to the power of 5. This operation is based on the principle that raising a root to its corresponding power will cancel out the root, leaving us with the radicand. Raising both sides to the power of 5 is a direct application of the power property of equality. This step is crucial because it directly eliminates the radical, allowing us to solve for the variable D. By raising the fifth root to the power of 5, we are essentially undoing the root operation and bringing D out from under the radical sign. This simplification is a pivotal moment in solving the equation, as it transforms the problem into a straightforward algebraic form.

  • Current Equation: √[5]{D} = -5
  • Raise both sides to the power of 5: (√[5]{D})^5 = (-5)^5
  • Simplified: D = -3125

Step 4: The Solution

After performing the necessary algebraic manipulations, we have arrived at the solution for D. From the previous step, we found that D = -3125. This is the value of D that satisfies the original equation, -11√[5]{D} - 11 = 44. To ensure the accuracy of our solution, it's always a good practice to substitute this value back into the original equation and verify that it holds true. This verification process is a crucial step in problem-solving, as it helps to catch any potential errors made during the algebraic manipulations. The solution D = -3125 represents the culmination of our step-by-step approach, where we carefully isolated the radical term, eliminated the coefficient, and finally, removed the fifth root to solve for D. This process demonstrates the power of algebraic techniques in unraveling complex equations and arriving at precise solutions.

  • Solution: D = -3125

Conclusion

In conclusion, we have successfully solved the equation -11√[5]{D} - 11 = 44 for the variable D. By following a systematic, step-by-step approach, we were able to isolate the radical term, eliminate the coefficient, and ultimately remove the fifth root to find the value of D. The solution we arrived at is D = -3125. Throughout this process, we emphasized the importance of understanding and applying key algebraic principles, such as the addition, division, and power properties of equality. These principles are fundamental to manipulating equations and arriving at correct solutions. Furthermore, we highlighted the significance of verifying the solution by substituting it back into the original equation, ensuring the accuracy of our work. Solving equations involving radicals can seem challenging at first, but with practice and a clear understanding of the underlying concepts, it becomes a manageable task. The skills and knowledge gained from solving this equation can be applied to a wide range of algebraic problems, making it a valuable exercise in mathematical problem-solving. This methodical approach not only helps in finding the correct answer but also in building a solid foundation in algebra, which is essential for more advanced mathematical studies. The key takeaway is that complex equations can be broken down into simpler steps, making them easier to solve. Each step, from isolating the radical to applying inverse operations, plays a crucial role in the overall solution process.