\( \left. \begin{array} { l l } { 2.1 } & { 2 ^ { 7 } \div 2 ^ { 4 } } \\ { 2.2 } & { \frac { 15 x ^ { 4 } y ^ { 2 } } { 5 x ^ { 3 } y ^ { 3 } } } \\ { 2.3 } & { 12 x ^ { 7 } \div 2 x ^ { 3 } } \end{array} \right. \)

The expression \( 2^7 \div 2^4 \) simply requires using the laws of exponents. You can subtract the exponents since the bases are the same: \( 2^{7-4} = 2^3 \). This gives you \( 2^3 = 8 \). A little math magic turns division into subtraction! For \( \frac{15 x^4 y^2}{5 x^3 y^3} \), start by simplifying the coefficients: \( 15 \div 5 = 3 \). Now handle the variables separately: for \( x \), subtract the exponents \( 4 - 3 = 1 \) (so it’s just \( x \)), and for \( y \), subtract \( 2 - 3 = -1 \) (yielding \( \frac{1}{y} \)). In the end, you get \( 3xy^{-1} \) or simply \( \frac{3x}{y} \)! Lastly, \( 12x^7 \div 2x^3 \) starts with the coefficients as \( 12 \div 2 = 6 \). For the \( x \) variable, subtract the exponents \( 7 - 3 = 4 \), and you write it all together as \( 6x^4 \). So, with a bit of fraction finesse, division of polynomials becomes easy-peasy!