Questions: Match the given equation with the correct first step for solving it. ∛[x(x+4)]=∛[-6] Choose the correct answer below. A. Let u=(x+4)^(1/5) and u^2=(x+4)^(2/5) B. Raise each side of the equation to the power 2/5. C. Multiply each side of the equation by x(x+4). D. Cube each side of the equation. E. Square each side of the equation.

Solution Steps

To solve the given equation \(\sqrt[3]{x(x+4)} = \sqrt[3]{-6}\), the first step is to eliminate the cube roots by cubing both sides of the equation. This will simplify the equation to a polynomial form that can be solved for \(x\).

1. Cube both sides of the equation to eliminate the cube roots.
2. Simplify the resulting polynomial equation.
3. Solve for \(x\).

To eliminate the cube roots, we cube both sides of the equation: \[ \left( \sqrt[3]{x(x+4)} \right)^3 = \left( \sqrt[3]{-6} \right)^3 \] This simplifies to: \[ x(x+4) = -6 \]

Next, we simplify the polynomial equation: \[ x^2 + 4x = -6 \] Rearranging terms, we get: \[ x^2 + 4x + 6 = 0 \]

We solve the quadratic equation \(x^2 + 4x + 6 = 0\) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 1\), \(b = 4\), and \(c = 6\). Substituting these values in, we get: \[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} = \frac{-4 \pm \sqrt{16 - 24}}{2} = \frac{-4 \pm \sqrt{-8}}{2} \] Since \(\sqrt{-8} = 2i\sqrt{2}\), we have: \[ x = \frac{-4 \pm 2i\sqrt{2}}{2} = -2 \pm i\sqrt{2} \]

Final Answer