Which Step Solves √[4](2x) + √[4](x+3) = 0? A Detailed Explanation

Let's embark on a journey to solve the intriguing equation $\sqrt[4]{2x} + \sqrt[4]{x+3} = 0$. This equation involves radicals, specifically fourth roots, which adds a layer of complexity. Our mission is to isolate the variable x and determine its value, while ensuring we follow valid algebraic steps. To achieve this, we'll dissect the equation, explore different approaches, and identify the correct first step towards the solution. This comprehensive guide will not only provide the answer but also illuminate the underlying principles of solving radical equations.

Understanding the Equation's Structure

Before diving into specific solution steps, it's crucial to understand the equation's structure. We are presented with an equation where two fourth roots are added together, and the sum equals zero. This immediately suggests a strategy: isolating one of the radicals on one side of the equation. By doing so, we can then raise both sides to the power of four, effectively eliminating the radical. However, a critical consideration arises – the domain of the radicals. Since we are dealing with fourth roots, the expressions under the radicals must be non-negative. This means both $2x \geq 0$ and $x+3 \geq 0$, which implies $x \geq 0$ and $x \geq -3$. The intersection of these conditions is $x \geq 0$. This constraint is crucial; any solution we find must satisfy this condition to be valid. Now, let's consider the provided options and analyze each one in the context of these principles.

Analyzing the Proposed Steps

The heart of solving any equation lies in performing valid algebraic manipulations. Each step must maintain the equality, meaning any operation performed on one side must also be performed on the other. With radical equations, squaring or raising to higher powers is a common technique to eliminate the radicals. However, this process can sometimes introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. Therefore, checking solutions at the end is always a necessity. Now, let's examine the given options and determine which one represents a valid initial step.

Option A: $(\sqrt[4]{2x})^4 = (\sqrt[4]{x+3})^4$

This option suggests raising both sides of an equation to the power of four. But before we can apply this operation, we need to isolate the radical terms. Directly raising the original equation $\sqrt[4]{2x} + \sqrt[4]{x+3} = 0$ to the fourth power would be a complex undertaking due to the cross terms that would arise from the binomial expansion. The correct approach involves first isolating one radical on one side of the equation. So, this option, in its current form, isn't the immediate valid step, as it skips the crucial isolation stage. While raising to the fourth power is a valid operation, applying it directly at this stage is premature.

Option B: $(\sqrt[4]{2x})^2 = (\sqrt[4]{x+3})^2$

Similar to Option A, this option suggests squaring both sides. Squaring can eliminate square roots, but here, we have fourth roots. Squaring a fourth root results in a square root (e.g., $(\sqrt[4]{a})^2 = \sqrt{a}$). Again, the fundamental issue is the absence of isolation. Applying this step directly to the original equation wouldn't simplify the problem effectively. We would still have radicals to contend with. Furthermore, just like Option A, it is not the immediate valid step, as it skips the crucial isolation stage.

Option C: $(\sqrt[4]{2x})^2 = (-\sqrt[4]{x+3})^2$

This option introduces a subtle but crucial difference. To see if this option is a valid step, we first need to rewrite the original equation by subtracting $\sqrt[4]{x+3}$ from both sides. This gives us $\sqrt[4]{2x} = -\sqrt[4]{x+3}$. Now, squaring both sides of this isolated equation gives us $(\sqrt[4]{2x})^2 = (-\sqrt[4]{x+3})^2$. This step is valid because it follows from a correct isolation of terms and a valid algebraic manipulation (squaring both sides). Remember that squaring a negative term results in a positive term, so the negative sign on the right side is accounted for. It accurately represents a step in the solution process. This is a significant step towards simplification.

Option D: $(\sqrt[4]{2x})^4 = (\sqrt[4]{x-3})^4$

Option D presents two critical flaws. First, it involves a sign error. To isolate the radicals, we should subtract $\sqrt[4]{x+3}$ from both sides, leading to a negative sign on one side. However, this option doesn't account for that. Second, it changes the original equation by replacing "x+3" with "x-3" under the radical. This alteration fundamentally changes the equation and makes this step invalid. Therefore, Option D is incorrect and represents a significant departure from the correct solution path. It's crucial to maintain the integrity of the equation throughout the solution process, and this option fails to do so.

The Correct First Step: Option C in Detail

Option C, $(\sqrt[4]{2x})^2 = (-\sqrt[4]{x+3})^2$, stands out as the valid first step. Let's delve deeper into why and how this step propels us towards the solution.

1. Isolating the Radicals: The initial move is to isolate the radical terms. Starting with $\sqrt[4]{2x} + \sqrt[4]{x+3} = 0$, we subtract $\sqrt[4]{x+3}$ from both sides, yielding $\sqrt[4]{2x} = -\sqrt[4]{x+3}$. This isolation is paramount because it allows us to apply the squaring operation effectively.
2. Squaring Both Sides: Now, with the radicals isolated, we can square both sides of the equation. Squaring both sides of $\sqrt[4]{2x} = -\sqrt[4]{x+3}$ gives us $(\sqrt[4]{2x})^2 = (-\sqrt[4]{x+3})^2$. This step is mathematically sound and simplifies the radicals, reducing the fourth roots to square roots.
3. Simplifying the Equation: Squaring the terms simplifies the equation to $\sqrt{2x} = \sqrt{x+3}$. The negative sign on the right-hand side vanishes because squaring a negative term results in a positive term. This simplification is a crucial step forward in the solution process.

By correctly isolating the radicals and then squaring, Option C sets us on the right path to solving the equation. It demonstrates an understanding of the principles of algebraic manipulation and the nature of radical equations. This step transforms the equation into a more manageable form, paving the way for further simplification and the eventual solution.

Continuing the Solution

Now that we've identified Option C as the correct first step, let's briefly outline the subsequent steps required to solve the equation completely. From the simplified equation $\sqrt{2x} = \sqrt{x+3}$, we can square both sides again to eliminate the remaining square roots. This yields $2x = x + 3$. Solving this linear equation is straightforward: subtract x from both sides to get $x = 3$.

However, we must remember the crucial step of checking for extraneous solutions. We substitute $x = 3$ back into the original equation: $\sqrt[4]{2(3)} + \sqrt[4]{3+3} = \sqrt[4]{6} + \sqrt[4]{6}$. This clearly does not equal zero. This might lead us to believe that there is an error, but let's revisit the step where we isolated the radicals. The equation $\sqrt[4]{2x} = -\sqrt[4]{x+3}$ implies that a positive quantity (the fourth root of 2x) equals a negative quantity (the negative of the fourth root of x+3). This can only be true if both terms are zero. So, we need to solve the system of equations $2x = 0$ and $x+3 = 0$. These equations give us $x = 0$ and $x = -3$, respectively. However, a value of x cannot simultaneously be 0 and -3, therefore, the original equation has no solution.

Conclusion

In conclusion, the valid first step in solving the equation $\sqrt[4]{2x} + \sqrt[4]{x+3} = 0$ is Option C, $(\sqrt[4]{2x})^2 = (-\sqrt[4]{x+3})^2$. This step correctly follows from isolating the radical terms and applying a valid algebraic manipulation (squaring both sides). While the subsequent steps lead to a potential solution, checking for extraneous solutions reveals that the original equation has no solution. This comprehensive exploration highlights the importance of understanding the structure of equations, applying valid algebraic steps, and rigorously verifying solutions in the context of radical equations.

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Which of the following is a valid first step in solving the equation $\sqrt[4]{2x} + \sqrt[4]{x+3} = 0$?

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Solving $\sqrt[4]{2x} + \sqrt[4]{x+3} = 0$ A Step-by-Step Guide and Solution