$\text{a)}{\left(4+2\sqrt{2}\right)}^{\frac{2}{3}}\cdot {\left(4-2\sqrt{2}\right)}^{\frac{2}{3}}={\left(4^2-{\left(2\sqrt{2}\right)}^2\right)}^{\frac{2}{3}}={\left(16-8\right)}^{\frac{2}{3}}={\left(8\right)}^{\frac{2}{3}}=\sqrt[3]{64}=4$  

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$\text{b)}{\left({\left(4+7^{\frac{1}{2}}\right)}^{\frac{1}{2}}+{\left(4-7^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}^2=4+7^{\frac{1}{2}}+2\cdot {\left(4+7^{\frac{1}{2}}\right)}^{\frac{1}{2}}\cdot {\left(4-7^{\frac{1}{2}}\right)}^{\frac{1}{2}}+4-7^{\frac{1}{2}}=$  

$=8+2\cdot \left({\left(16-7\right)}^{\frac{1}{2}}\right)=8+2\cdot 9^{\frac{1}{2}}=8+2\cdot 3=8+6=14$