Solve the cube root of (5 - 3x) equals 4.

Step-by-step solution

1. 1

   Notice that a cube root behaves differently from a square root. The function $t\mapsto\sqrt[3]{t}$ is defined for every real $t$ and is strictly increasing, so there is no domain restriction and no possibility of extraneous solutions. No sign condition is needed on either side.

2. 2

   Cube both sides. Cubing is the exact inverse of the cube root: $5-3x = 4^3 = 64.$

3. 3

   Isolate the x-term. $-3x = 64-5 = 59.$

4. 4

   Divide by -3. $x = -\frac{59}{3} \approx -19.667.$ Keep the fraction exact; $59$ is prime, so it does not reduce.

5. 5

   Verify by substitution. $5-3\left(-\tfrac{59}{3}\right) = 5+59 = 64$, and $\sqrt[3]{64} = 4$, matching the right-hand side.

6. 6

   Contrast with the square-root case. If the equation had been $\sqrt{5-3x} = 4$ the same algebra would apply, but the radicand would additionally have to be nonnegative and the right side positive. With an odd-order root neither check exists, and the solution is always unique.