#### Answer

$a)$ $3\sqrt{3a^{2}+7a}$
$b)$ $5\sqrt{3t+4t^{2}}$

#### Work Step by Step

$a)$ $\sqrt{27a^{2}+63a}$
Take out common factor $9$ from the expression inside the square root:
$\sqrt{27a^{2}+63a}=\sqrt{9(3a^{2}+7a)}=...$
Take the square root of each factor:
$...=(\sqrt{9})(\sqrt{3a^{2}+7a})=3\sqrt{3a^{2}+7a}$
$b)$ $\sqrt{75t+100t^{2}}$
Take out common factor $25$ from the expression inside the square root:
$\sqrt{75t+100t^{2}}=\sqrt{25(3t+4t^{2})}=...$
Take the square root of each factor:
$...=(\sqrt{25})(\sqrt{3t+4t^{2}})=5\sqrt{3t+4t^{2}}$