What is the solution of $\sqrt[3]{x+8}=-4$ ?
The given equation is \(\sqrt[3]{x+8} = -4\). We need to determine the value of \(x\) that satisfies this equality. Since the function \(f(t) = \sqrt[3]{t}\) is defined for all real numbers and is the inverse operation of cubing, we can isolate \(x\) by raising both sides of the equation to the power of 3. This operation is valid for cube roots without introducing extraneous solutions.
To eliminate the cube root on the left side, we raise both sides of the equation to the power of 3:
\[(\sqrt[3]{x+8})^3 = (-4)^3\]
The left side simplifies to \(x+8\), as cubing a cube root restores the original radicand. For the right side, we calculate \((-4) \times (-4) \times (-4)\):
\[(-4) \times (-4) = 16\]
\[16 \times (-4) = -64\]
Thus, the equation simplifies to:
\[x + 8 = -64\]
Now, we solve for \(x\) by subtracting 8 from both sides of the equation:
\[x = -64 - 8\]
\[x = -72\]
Finally, we verify the solution by substituting \(x = -72\) back into the original expression:
\[\sqrt[3]{-72 + 8} = \sqrt[3]{-64} = -4\]
Since the result matches the original equation, \(x = -72\) is the correct solution. This corresponds to the first option provided.
Cube roots are frequently encountered in geometry and physics when relating volume to length. For instance, if you have a cubic container with a known volume and you need to determine the length of its side, you calculate the cube root of the volume. Solving algebraic equations involving cube roots, like the one here, allows you to determine dimensions or unknown parameters in such physical systems, providing a foundation for understanding inverse operations and power functions.