In two or more complete sentences, explain how to solve the cube root equation, $\sqrt[3]{x-1}+2=0$.
Answer (2)
To solve the equation 3 x − 1 + 2 = 0 , isolate the cube root by subtracting 2, then cube both sides to eliminate the cube root. Finally, solve for x to find that the solution is x = − 7 .
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Isolate the cube root term: 3 x − 1 = − 2 .
Cube both sides: ( 3 x − 1 ) 3 = ( − 2 ) 3 .
Simplify: x − 1 = − 8 .
Solve for x : x = − 7 , so the final answer is − 7 .
Explanation
Isolating the Cube Root We are given the equation 3 x − 1 + 2 = 0 and we want to find the value of x that satisfies this equation. Our first goal is to isolate the cube root term on one side of the equation.
Subtracting 2 from Both Sides To isolate the cube root, we subtract 2 from both sides of the equation: 3 x − 1 + 2 − 2 = 0 − 2 3 x − 1 = − 2
Cubing Both Sides Now that we have isolated the cube root, we can eliminate it by cubing both sides of the equation: ( 3 x − 1 ) 3 = ( − 2 ) 3
Simplifying the Equation Cubing a cube root cancels out, and ( − 2 ) 3 = − 2 × − 2 × − 2 = − 8 , so we have: x − 1 = − 8
Adding 1 to Both Sides To solve for x , we add 1 to both sides of the equation: x − 1 + 1 = − 8 + 1 x = − 7
Final Answer Therefore, the solution to the equation 3 x − 1 + 2 = 0 is x = − 7 .
Examples
Cube root equations can be used to model various real-world phenomena, such as determining the side length of a cube given its volume. For example, if you have a cube-shaped container and you know its volume is 125 cubic inches, you can use a cube root to find the length of one side. The equation would be s = 3 125 , where s is the side length. Solving this, you find that each side is 5 inches long. Cube roots are also used in fields like acoustics and fluid dynamics.
