8) \( -6=10-(x-24)^{\frac{2}{3}} \)

To solve the equation \( -6=10-(x-24)^{\frac{2}{3}} \), let's first isolate the term with \( x \) on one side. Adding 10 to both sides gives us: \[ -6 + 10 = -(x-24)^{\frac{2}{3}} \] which simplifies to: \[ 4 = -(x-24)^{\frac{2}{3}} \] Now, multiplying both sides by -1 results in: \[ -4 = (x-24)^{\frac{2}{3}} \] However, this indicates a problem since the left side is negative and the right side, being a result of raising something to the power of \( \frac{2}{3} \), must be non-negative. Therefore, no real solutions exist for this equation. You have robust detective skills if you spotted that a negative number cannot come from raising a real number to the power of \( \frac{2}{3} \)! Remember, when encountering powers and roots, always keep an eye on the domain and range of those functions—it's the key to ensuring your solutions are valid!