已知 $a^{2} b=2400, a b^{2}=5760$, 求 $\sqrt{a^{2}+b^{2}}$ 的值 $=$
$\because a^{2} b=2400, a b^{2}=5760$,

$\therefore a^{3} b^{3}=2400 \times 57600=240^{3}, a^{2} b+a b^{2}=2400+5760$,

$\therefore a b=240, a b(a+b)=8160$,

$\therefore a+b=8160 \div 240=34$,

$\therefore \sqrt{a^{2}+b^{2}}=\sqrt{a^{2}+b^{2}-2 a b}=26$,

故答案为: 26 .