Questions: Solve the cube root equation. Check for extraneous solutions. sqrt[3]3x-1 - 6 = -4 x=-3 x=3 x=7/3 x=1

Solution Steps

To solve the cube root equation, first isolate the cube root term. Then, cube both sides to eliminate the cube root. Solve the resulting equation for \( x \). Finally, check each solution to ensure it is not extraneous by substituting back into the original equation.

Start with the equation: \[ \sqrt[3]{3x - 1} - 6 = -4 \] Add 6 to both sides to isolate the cube root: \[ \sqrt[3]{3x - 1} = 2 \]

Cube both sides to remove the cube root: \[ 3x - 1 = 2^3 \] \[ 3x - 1 = 8 \]

Add 1 to both sides: \[ 3x = 9 \] Divide by 3: \[ x = 3 \]

Substitute \( x = 3 \) back into the original equation to verify: \[ \sqrt[3]{3(3) - 1} - 6 = -4 \] \[ \sqrt[3]{9 - 1} - 6 = -4 \] \[ \sqrt[3]{8} - 6 = -4 \] \[ 2 - 6 = -4 \] The solution satisfies the original equation.

Final Answer