Solving The Equation $\sqrt[5]{x+7}=-2$ A Step-by-Step Guide
The given equation is $\sqrt[5]{x+7}=-2$. To solve for x, we first need to isolate the radical term. In this case, the radical term is already isolated on the left side of the equation. This is a crucial first step in dealing with equations involving radicals because it sets the stage for eliminating the radical by raising both sides to an appropriate power. When we isolate the radical, we ensure that the operation we perform on both sides will effectively remove the radical sign, allowing us to work directly with the expression inside the radical. In more complex equations, this might involve adding or subtracting terms to move them to the other side or dividing to eliminate coefficients multiplying the radical. However, in this instance, the equation is straightforward, and the radical is already by itself, which simplifies our task significantly. The next step will involve using the properties of exponents and roots to undo the fifth root, which will bring us closer to finding the value of x that satisfies the original equation. By carefully following these steps, we can systematically solve the equation and arrive at the correct answer. Isolating the radical not only simplifies the process but also helps in avoiding common mistakes that can occur when dealing with radical equations.
After isolating the radical, the next crucial step in solving the equation $\sqrt[5]{x+7}=-2$ is to eliminate it. Since we have a fifth root (an index of 5), we need to raise both sides of the equation to the power of 5. This is based on the property that $(am)n = a^{m*n}$ and that the nth power of an nth root will cancel out, leaving the radicand. Therefore, we raise both sides of the equation to the power of 5:
On the left side, raising the fifth root to the power of 5 cancels out the root, leaving us with x + 7. On the right side, we calculate (-2)^5, which means multiplying -2 by itself five times:
So, after raising both sides to the power of 5, the equation becomes:
This step is essential because it transforms the original radical equation into a simpler linear equation, which is much easier to solve. By understanding and correctly applying the properties of exponents and roots, we efficiently eliminate the radical, paving the way for the final steps in finding the solution for x. This careful manipulation ensures that we maintain the equality while simplifying the equation.
Now that we've eliminated the radical, we have a simple linear equation: x + 7 = -32. The next step is to isolate x on one side of the equation. To do this, we subtract 7 from both sides of the equation. This maintains the equality and moves us closer to finding the value of x:
This simplifies to:
So, we have found a potential solution for x, which is -39. It's important to remember that with radical equations, we must check our solution to ensure it doesn't introduce any extraneous roots. Extraneous roots are solutions that arise during the solving process but do not satisfy the original equation. This often happens because raising both sides of an equation to an even power can introduce solutions that weren't there originally. Therefore, it's crucial to verify whether our solution, x = -39, is valid by plugging it back into the original equation. By following these algebraic steps carefully, we've successfully isolated x and found its value, but the journey isn't complete until we've confirmed its validity in the original equation.
After finding a potential solution for x in the equation $\sqrt[5]{x+7}=-2$, the most critical step is to check whether this solution is valid. This is particularly important in equations involving radicals because raising both sides to a power can sometimes introduce extraneous solutions—values that satisfy the transformed equation but not the original one. Our potential solution is x = -39. To check this, we substitute -39 back into the original equation:
Simplify the expression inside the radical:
Now, we need to determine if the fifth root of -32 is indeed -2. The fifth root of a number is a value that, when raised to the power of 5, equals the number. In this case, we are asking if there is a number that, when multiplied by itself five times, equals -32. We know that:
So, the fifth root of -32 is indeed -2. Therefore, our equation becomes:
Since this statement is true, our solution x = -39 is valid. This check confirms that -39 is a legitimate solution to the original equation and not an extraneous root introduced during the solving process. By meticulously checking our solution, we ensure the accuracy and correctness of our result.
After carefully solving the equation $\sqrt[5]{x+7}=-2$ and verifying our solution, we have arrived at the final answer. We began by isolating the radical, then eliminated it by raising both sides of the equation to the power of 5. This transformed the radical equation into a simple linear equation, which we solved for x. We found a potential solution of x = -39. However, the crucial step in dealing with radical equations is to check this solution in the original equation to ensure it's valid and not an extraneous root.
When we substituted x = -39 back into the original equation, we found:
Since this equality holds true, we confirmed that x = -39 is indeed a valid solution.
Therefore, the correct answer to the equation $\sqrt[5]{x+7}=-2$ is:
A. -39
This methodical approach, from isolating the radical to verifying the solution, ensures that we avoid common pitfalls in solving radical equations and arrive at the accurate result. By understanding the properties of radicals and exponents, and by meticulously checking our work, we can confidently solve these types of equations.
