Find the real solutions of the equation (2x + 2)^(1/3) = −4.

Step-by-step solution

1. 1

   Read the exponent as a cube root. By definition $a^{1/3} = \sqrt[3]{a}$, so the equation says: the cube root of $2x+2$ equals $-4$. A negative value on the right is legitimate here — every real number has exactly one real cube root, including negatives.

2. 2

   Cube both sides to undo the root. Raising both sides to the third power is the inverse of taking a cube root:

   $\left((2x+2)^{1/3}\right)^3 = (-4)^3 \quad\Longrightarrow\quad 2x + 2 = -64$

   Note $(-4)^3 = -64$, keeping the sign, unlike squaring which would destroy it.

3. 3

   Solve the resulting linear equation.

   $2x = -64 - 2 = -66 \quad\Longrightarrow\quad x = -33$

4. 4

   Verify in the original equation. Substituting $x = -33$ gives $2(-33) + 2 = -64$, and $(-64)^{1/3} = -4$, which matches the right-hand side exactly.

5. 5

   Why no extraneous root check is needed. Cubing is a one-to-one operation on the real numbers, so it neither gains nor loses solutions. That is the key difference from square-root equations, where squaring both sides can invent solutions that must be discarded. Here $x = -33$ is the only real solution.