$a\text{)}{2}^{\sqrt{2}}\cdot 2=2^{\sqrt{2}+1}$  

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$b\text{)}{2}^{\sqrt{2}}\cdot 2^{\sqrt{2}}=2^{\sqrt{2}+\sqrt{2}}=2^{2\sqrt{2}}$  

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$c\text{)}{2}^{1-\sqrt{2}}\cdot 2^{1+\sqrt{2}}=2^{1-\sqrt{2}+1+\sqrt{2}}=2^2$  

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$d\text{)}{\left(\frac{1}{4}\cdot 2^{\sqrt{2}}\right)}^{\sqrt{2}}={\left({\left(\frac{1}{2}\right)}^2\cdot 2^{\sqrt{2}}\right)}^{\sqrt{2}}={\left({\left(2^{-1}\right)}^2\cdot 2^{\sqrt{2}}\right)}^{\sqrt{2}}=$  

$={\left(2^{-2}\cdot 2^{\sqrt{2}}\right)}^{\sqrt{2}}={\left(2^{-2+\sqrt{2}}\right)}^{\sqrt{2}}=2^{2-2\sqrt{2}}$  

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$e\text{)}\frac{2^{\sqrt{2}}}{2}=2^{\sqrt{2}-1}$  

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$f\text{)}{2}^{\sqrt{2}}\cdot \sqrt{2}=2^{\sqrt{2}}\cdot 2^{\frac{1}{2}}=2^{\sqrt{2}+\frac{1}{2}}$  

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$g\text{)}\frac{8^{\sqrt{2}}}{2^{\sqrt{2}}}=\frac{{\left(2^3\right)}^{\sqrt{2}}}{2^{\sqrt{2}}}=\frac{2^{3\sqrt{2}}}{2^{\sqrt{2}}}=2^{3\sqrt{2}-\sqrt{2}}=2^{2\sqrt{2}}$  

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$h\text{)}{\left(\frac{16}{2^{\sqrt{2}}}\right)}^{\sqrt{2}}=\frac{{\left(2^4\right)}^{\sqrt{2}}}{2^{\sqrt{2}\cdot \sqrt{2}}}=\frac{2^{4\sqrt{2}}}{2^2}=2^{4\sqrt{2}-2}$