Select the correct answer.
What is the solution to the equation?
$\sqrt[5]{x+7}=-2$
We are tasked with solving the equation \(\sqrt[5]{x+7} = -2\) for \(x\). The expression involves a fifth root of a linear polynomial \(x+7\). Since the index of the radical is odd (5), the domain of the function is all real numbers, and the range is also all real numbers. Thus, we expect a unique real solution for \(x\).
To isolate \(x\), we raise both sides of the equation to the power of 5:
\[(\sqrt[5]{x+7})^5 = (-2)^5\]
Applying the property \((\sqrt[n]{u})^n = u\) for any real \(u\) when \(n\) is odd, the left side simplifies:
\[x + 7 = (-2)^5\]
Now, calculate the right side:
\[(-2)^5 = -2 \times -2 \times -2 \times -2 \times -2 = -32\]
So, the equation becomes:
\[x + 7 = -32\]
To solve for \(x\), we subtract 7 from both sides of the equation:
\[x = -32 - 7\]
\[x = -39\]
Finally, we verify the solution by substituting \(x = -39\) into the original equation:
\[\sqrt[5]{-39 + 7} = \sqrt[5]{-32} = -2\]
The calculation is correct. Comparing this result with the provided options, we conclude that the correct solution is \(-39\). Thus, the correct option is \(-39\).
In fields like engineering or physics, we often deal with inverse functions to solve for unknown variables. For example, if the displacement of an object follows the relationship \(s = \sqrt[5]{t+7}\), and we observe a displacement of \(-2\) units (which might represent position relative to a reference point in a specialized system), we would need to solve for the time \(t\). This method of applying inverse powers allows us to isolate the variable of interest, providing a clear pathway from observations to initial conditions.