For relatively small values of $n,$ algorithms with larger orders can be more efficient than algorithms with smaller orders.

Suppose that when run with an input of size $n,$ algorithm A requires ${10}^7{n}^2$ operations and algorithm $B$ requires $n^3$ operations.

$($a$)$ What are orders for algorithms A and $B$ from among the set of power functions?

algorithm A order

algorithm $B$ order

$($b$)$ For what values of $n$ is algorithm A more efficient than algorithm $B?($Enter your answer as a single inequality solved for n$.)$

$($c$)$ For what values of $n$ is algorithm B at least $100$ times more efficient than algorithm A$?($Enter your answer as a single inequality solved for $n.)$