$a)\sqrt{11-\sqrt{21}}\cdot \sqrt{11+\sqrt{21}}=\sqrt{\left(11-\sqrt{21}\right)\left(11+\sqrt{21}\right)}=$

$=\sqrt{11\cdot 11+11\cdot \sqrt{21}-\sqrt{21}\cdot 11-{\left(\sqrt{21}\right)}^2}=\sqrt{121-21}=\sqrt{100}=10$   

$b)\sqrt[3]{7+\sqrt{22}}-\sqrt[3]{7-\sqrt{22}}=\sqrt[3]{\left(7+\sqrt{22}\right)\left(7-\sqrt{22}\right)}=$    

$=\sqrt[3]{7\cdot 7-7\cdot \sqrt{22}+\sqrt{22}\cdot 7-{\left(\sqrt{22}\right)}^2}=\sqrt[3]{49-22}=\sqrt[3]{27}=3$