Final answer:
The expression square root of (-5x)^3 makes sense for non-negative values of x, including zero, as the cubic term will maintain its sign, and the square root of a negative number is not defined in the real number system.
Step-by-step explanation:
The question asks for which values of x the expression square root of (-5x)^3 makes sense.
Since we're looking at the cube of -5x, and any real number (positive, negative, or zero) raised to an odd power will still result in a real number, the cubic term is not the issue.
However, the square root is only real for non-negative values, which requires the inside of the square root to be greater than or equal to zero.
Consider the expression inside the square root: (-5x)^3. The cube of any real number will maintain its sign, so if x is negative, the cube will be negative, and taking the square root of a negative number is not possible in the real number system.
Therefore, x must be non-negative in order for the expression to make sense.
But if x is 0, the cube would be 0, and the square root of 0 is 0, which does make sense. Thus, the only restriction is that x cannot be negative.
So the values of x for which the expression makes sense are all non-negative numbers, including zero.