Find the Domain

Problem

3√(3,(m^8)/64)−5*m^2√(3,(m^2)/27)

Solution

1. Identify the type of function involved in the expression. The expression consists of cube roots and polynomial terms.

2. Recall the properties of radical functions with odd indices. The cube root function, √(3,x) is defined for all real numbers, including positive numbers, negative numbers, and zero.

3. Analyze the radicands (the expressions inside the roots). The radicands are (m^8)/64 and (m^2)/27 Both are defined for all real values of m because they are simple power functions divided by constants.

4. Determine if there are any restrictions. Since there are no denominators containing variables and no even-indexed roots, there are no values of m that would make the expression undefined.

5. Conclude that the domain is the set of all real numbers.

Final Answer

Domain=(−∞,∞)

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