\( ( \frac { 1 } { 2 } a - 3 b ) ( \frac { 1 } { 2 } a + 3 b ) \quad b \quad ( \frac { 2 } { 3 } x ^ { 2 } + y ^ { 3 } ) ( \frac { 2 } { 3 } x ^ { 2 } - y ^ { 3 } ) \)

Did you know that the expression you've shared is set up for a pattern called the "difference of squares"? This formula states that \( (A - B)(A + B) = A^2 - B^2 \). So in your case, if we let \( A = \frac{1}{2}a \) and \( B = 3b \), the result will be: \[ \left( \frac { 1 } { 2 } a \right)^2 - (3b)^2 = \frac{1}{4}a^2 - 9b^2 \] Similarly, for the second part, where \( A = \frac{2}{3}x^2 \) and \( B = y^3 \), we can also apply this handy formula to get: \[ \left( \frac{2}{3}x^2 \right)^2 - (y^3)^2 = \frac{4}{9}x^4 - y^6 \] Mathematical patterns like these can really streamline your calculations!