What is the solution to this equation?
$7-\sqrt[3]{2-x}=12$
The solution is $x=$ ___.
The objective is to find the value of \(x\) that satisfies the equation \(7 - \sqrt[3]{2 - x} = 12\). We observe that the variable \(x\) is contained within a cube root. To isolate \(x\), we need to manipulate the equation algebraically by isolating the radical term first, then eliminating the cube root by cubing both sides of the equation. This approach will allow us to convert the radical equation into a simple linear form.
Start by isolating the radical term \(\sqrt[3]{2 - x}\) on one side of the equation. Subtract \(7\) from both sides:
\[-\sqrt[3]{2 - x} = 12 - 7\]
\[-\sqrt[3]{2 - x} = 5\]
Next, multiply both sides by \(-1\) to clear the negative sign:
\[\sqrt[3]{2 - x} = -5\]
To eliminate the cube root, raise both sides of the equation to the power of \(3\):
\[(\sqrt[3]{2 - x})^3 = (-5)^3\]
This simplifies the equation to:
\[2 - x = -125\]
Now, solve for \(x\) by rearranging the terms:
\[x = 2 + 125\]
\[x = 127\]
Verify the result by substituting \(x = 127\) back into the original equation:
\[7 - \sqrt[3]{2 - 127} = 7 - \sqrt[3]{-125}\]
Since \(\sqrt[3]{-125} = -5\), the expression becomes:
\[7 - (-5) = 7 + 5 = 12\]
The value satisfies the original equation, so the solution is \(x = 127\). The final answer is \(\boxed{127}\).
Radical equations like this one appear in fields such as engineering and physics, where relationships between physical quantities often involve powers or roots. For example, if the velocity of an object in a specific medium is related to the cube root of the difference between pressure and a reference point, one could use this algebraic method to determine the specific pressure needed to reach a desired velocity. Mastering these techniques allows for precise modeling of physical systems and efficient problem-solving in science and technology.