((3x^((2)/(2))y^(3))/(x^(2)y^(-(1)/(2))))^(-2)

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Simplify Inside Parentheses: First, simplify the expression inside the parentheses before applying the exponent of -2. The expression x^(2/2) simplifies to x^(1) or just x, because any number raised to the power of 1 is itself.Divide Like Bases: Now, simplify the expression inside the parentheses by dividing the terms with the same base. (3x^1y^3) / (x^2y^(-1/2)) simplifies to 3x^(1-2)y^(3-(-1/2)) by subtracting the exponents of like bases.Subtract Exponents: Perform the subtraction of the exponents for x and y. For x: 1 - 2 = -1 For y: 3 - (-1/2) = 3 + 1/2 = 3.5 or 7/2 The simplified expression inside the parentheses is now 3x^(-1)y^(7/2).Apply Exponent: Apply the exponent of -2 to the simplified expression. (3x^(-1)y^(7/2))^(-2) means that each factor inside the parentheses is raised to the power of -2.Raise to Power: Raise each factor inside the parentheses to the power of -2. For the coefficient 3: (3)^(-2) = 1/(3^2) = 1/9 For x^(-1): (x^(-1))^(-2) = x^((-1)*(-2)) = x^2 For y^(7/2): (y^(7/2))^(-2) = y^((7/2)*(-2)) = y^(-7) The expression now looks like (1/9)x^2y^(-7).Final Simplified Form: The final simplified form of the expression is (1/9)x^2y^(-7). Since y^(-7) is in the denominator when moved out of the exponent, the final answer is (1/9)x^2/(y^7).

$\left(\frac{3x^{\left(\frac{2}{2}\right)}y^{3}}{x^{2}y^{-\left(\frac{1}{2}\right)}}\right)^{-2}$

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$\left(\frac{3x^{\left(\frac{2}{2}\right)}y^{3}}{x^{2}y^{-\left(\frac{1}{2}\right)}}\right)^{-2}$