Find the Domain square root of 6x square root of 3x^2

Problem

√(,6*x√(,3*x2))

Solution

1. Identify the conditions for the domain of a square root function. The expression inside any square root (the radicand) must be greater than or equal to zero.

2. Analyze the innermost square root √(,3*x2) The radicand 3*x2 must satisfy 3*x2≥0 Since x2 is always non-negative for all real numbers, this condition is satisfied for all x

3. Simplify the expression to analyze the outer square root. Note that √(,x2)=|x| The expression becomes √(,6*x⋅|x|√(,3))

4. Set up the inequality for the outer radicand. We require 6*x⋅|x|√(,3)≥0

5. Solve the inequality. Since 6 |x| and √(,3) are always non-negative, the sign of the product depends on x For the product to be non-negative, we must have x≥0

6. Combine the conditions. The innermost root allows all real numbers, but the outer root restricts x to non-negative values.

Final Answer

Domain:[0,∞)

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