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

Evaluate \( \left(\frac{3.5^{7}}{3.5^{6}}\right)^{2} \)

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

\( f(x)=2 x-1 \) and \( g(x)=3 x^{2} \) Find both \( f \circ g \) and \( g \circ f \), stating domain and range of each.

(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 \( 4 c(-a+b+c \)

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