Answer
(a) $p_1^{2e_1}p_2^{2e_2}\cdots p_k^{2e_k}$
(b) $42, (5880)^2$
(c) $231, (4158)^2$
Work Step by Step
(a) Given $a=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$, we have
$a^2=p_1^{2e_1}p_2^{2e_2}\cdots p_k^{2e_k}$
(b) To find the least positive integer $n$, let $2^5\cdot 3\cdot 5^2\cdot 7^3\cdot n=2^6\cdot 3^2\cdot 5^2\cdot 7^4$, we have:
$n=2\cdot 3\cdot 7=42$ and
$2^5\cdot 3\cdot 5^2\cdot 7^3\cdot n=(2^3\cdot 3\cdot 5\cdot 7^2)^2=(5880)^2$
(c) To find the least positive integer $m$, let $2^2\cdot 3^5\cdot 7\cdot 11\cdot m=2^2\cdot 3^6\cdot 7^2\cdot 11^2$, we have:
$m=3\cdot 7\cdot 11=231$ and
$2^2\cdot 3^5\cdot 7\cdot 11\cdot m=(2\cdot 3^3\cdot 7\cdot 11)^2=(4158)^2$