The ring \( \mathbb{Z} \) has no nontrivial idempotent elements, whereas \( \mathbb{Z} \times \mathbb{Z} \) has the idernpotent elements \( (1,0) \) and \( (0,1) \).

\( \frac { 1 } { r } x \frac { 1 - ( \frac { 1 } { r } ) ^ { n } } { 1 - \frac { 1 } { r } } \geq \frac { 9 r } { 100 } \Rightarrow 1 - \frac { 1 } { r ^ { n } } \geq \frac { 9 r } { 100 } \)

Multiply the bino (IIi) \( (x-4) \) and ( \( 2 x \)

\( \frac { 1 } { r } \times \frac { 1 - ( \frac { 1 } { r } ) ^ { n } } { 1 - \frac { 1 } { r } } \geq \frac { 9 r } { 100 } \Rightarrow 1 - \frac { 1 } { r ^ { n } } \geq \frac { 9 r } { 100 } \)

Choose an equivalent expression for \( \left(\left(\frac{2}{5}\right)^{4}\left(\frac{2}{5}\right)^{3}\right)^{2} \)

(c) Subtract: \( 31(1-4 m+5 n) \) from \( 4 I(10 n-3 m+21) \) (d) Subtract: \( 3 a(a+b+c)-2 b(a-b+c) \) from \( 4 c(-a+b+c \)