$a)$

${30}^2+{40}^2=c^2$

$900+1600=c^2$

$c^2=2500$

$c=\sqrt{2500}=50$

 

 

$b)$

${\left(\sqrt{5}\right)}^2+2^2=c^2$

$5+4=c^2$

$c=\sqrt{9}=3$

 

 

 

$c)$

${\left(2\sqrt{2}-1\right)}^2+{\left(2\sqrt{2}+1\right)}^2=c^2$

$\left(4\cdot 2-4\sqrt{2}+1\right)+\left(4\cdot 2+4\sqrt{2}+1\right)=c^2$

$9-4\sqrt{2}+9+4\sqrt{2}=c^2$

$c^2=18$

$c=\sqrt{18}=\sqrt{9\cdot 2}=3\sqrt{2}$

 

 

 

$d)$

${\left(\sqrt{5}-\sqrt{3}\right)}^2+{\left(\sqrt{5}+\sqrt{3}\right)}^2=c^2$

$\left(5-2\sqrt{5}\cdot \sqrt{3}+3\right)+\left(5+2\sqrt{5}\cdot \sqrt{3}+3\right)=c^2$

$8-2\sqrt{15}+8+2\sqrt{15}=c^2$

$16=c^2$

$c=\sqrt{16}=4$

 

 

 

$e)$

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

$4\cdot 2+4\cdot 2=c^2$

$c^2=16$

$c=\sqrt{16}=4$

 

 

 

$f)$

$a=2,b=\frac{1}{2}$

$2^2+{\left(\frac{1}{2}\right)}^2=c^2$

$4+\frac{1}{4}=c^2$

$c=\sqrt{4\frac{1}{4}}=\sqrt{\frac{17}{4}}=\frac{\sqrt{17}}{2}$