Solving $\sqrt[5]{x+7}=-2$ A Step-by-Step Guide

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Understanding the Problem

When tackling equations involving radicals, it's crucial to understand the properties of roots and exponents. In this particular case, we are presented with a fifth root equation: $\sqrt[5]{x+7}=-2$. Our mission is to isolate the variable x and determine its value, if any, that satisfies the given equation. The core concept at play here is the inverse relationship between radicals and exponents. To eliminate a radical, we raise both sides of the equation to the power that corresponds to the index of the radical. For a fifth root, this means raising both sides to the power of 5. However, before we proceed, it's essential to consider the nature of roots, especially when dealing with odd and even indices. Odd roots, like the fifth root in our equation, can yield both positive and negative results, while even roots typically only yield non-negative results in the real number system. This distinction is crucial as it directly impacts the possible solutions we might obtain. Therefore, as we embark on the process of solving this equation, we must remain mindful of the properties of fifth roots and the implications for the solution set. A solid grasp of these principles will guide us toward an accurate resolution of the equation and prevent any potential pitfalls along the way.

Step-by-Step Solution

To solve the equation $\sqrt[5]{x+7}=-2$, we will systematically eliminate the radical and isolate the variable x. This process involves a series of algebraic manipulations, each grounded in fundamental mathematical principles. Our first step is to raise both sides of the equation to the power of 5. This operation is justified by the property that if two quantities are equal, then raising them to the same power will preserve their equality. By raising both sides to the fifth power, we effectively eliminate the fifth root on the left-hand side, thereby simplifying the equation. However, it's crucial to remember that this step is reversible only if we consider both positive and negative roots. When dealing with odd roots, such as the fifth root, this is not a concern. We can proceed with confidence, knowing that we are not introducing any extraneous solutions at this stage. After raising both sides to the fifth power, we obtain a new equation that is free of radicals. This equation is typically a polynomial equation, which we can then solve using standard algebraic techniques. In our case, the resulting equation is relatively simple, allowing us to isolate x in a straightforward manner. By performing the necessary operations, such as subtracting 7 from both sides, we arrive at the value of x that satisfies the original equation. However, it's paramount to verify this solution by substituting it back into the original equation. This step is essential to ensure that our solution is valid and that no errors were introduced during the solving process. Only after this verification can we confidently assert that we have found the correct solution to the equation.

  1. Raise both sides to the power of 5: To eliminate the fifth root, we raise both sides of the equation to the power of 5:

    (x+75)5=(−2)5{(\sqrt[5]{x+7})^5=(-2)^5}

  2. Simplify: This simplifies to:

    x+7=−32{x+7=-32}

  3. Isolate x: Subtract 7 from both sides:

    x=−32−7{x=-32-7}

    x=−39{x=-39}

Verification

To ensure the solution is correct, it's essential to substitute the obtained value of x back into the original equation. This process, known as verification, serves as a critical safeguard against errors and extraneous solutions. By plugging the calculated value of x into the original equation, we can directly assess whether it satisfies the equation's conditions. If the substitution results in a true statement, we can confidently affirm that our solution is valid. However, if the substitution leads to a contradiction or an undefined expression, it indicates that our solution is either incorrect or extraneous. In the case of radical equations, verification is particularly important because the process of raising both sides to a power can sometimes introduce solutions that do not actually satisfy the original equation. These extraneous solutions arise from the fact that the squaring or raising to an even power can obscure the sign of the original expression. Therefore, by meticulously verifying our solution, we ensure that it is not merely a product of algebraic manipulation but also a genuine solution to the problem at hand. In our specific equation, we substitute x = -39 into the original equation and evaluate both sides. If the left-hand side equals the right-hand side, we can confidently conclude that x = -39 is indeed the correct solution. However, if the two sides do not match, it signals the presence of an error in our calculations, prompting us to re-examine our steps and identify the source of the discrepancy.

  • Substitute x = -39 into the original equation:

    −39+75=−2{\sqrt[5]{-39+7}=-2} −325=−2{\sqrt[5]{-32}=-2}

  • Check if the equation holds:

    Since $(-2)^5 = -32$, the fifth root of -32 is indeed -2.

    −2=−2{-2=-2}

Final Answer

After systematically solving the equation $\sqrt[5]x+7}=-2$ and diligently verifying our solution, we arrive at the definitive answer. Our step-by-step approach involved raising both sides of the equation to the power of 5 to eliminate the radical, simplifying the resulting expression, and isolating the variable x. The algebraic manipulations led us to the value x = -39 as a potential solution. However, it is the crucial step of verification that solidifies our confidence in this answer. By substituting x = -39 back into the original equation, we confirmed that it indeed satisfies the given condition. The left-hand side of the equation, $\sqrt[5]{-39+7}$, simplifies to $\sqrt[5]{-32}$, which evaluates to -2, matching the right-hand side of the equation. This conclusive verification leaves no room for doubt x = -39 is the sole solution to the equation $\sqrt[5]{x+7=-2$. Our journey through this problem underscores the importance of methodical problem-solving and the necessity of verification. While algebraic manipulations provide the means to arrive at a potential solution, it is the rigorous verification process that distinguishes a correct answer from a mere algebraic artifact. In summary, the final answer to the equation $\sqrt[5]{x+7}=-2$ is x = -39, a result firmly grounded in both algebraic derivation and empirical verification.

Therefore, the correct answer is:

  • A. -39