a)
[tex]\sqrt{2^{2}+\dfrac{5^{2} \cdot3^{2}}{4^{2}}}=\sqrt{25+\dfrac{25\cdot9}{16}}=\sqrt\dfrac{25\cdot16+225}{16}}=\dfrac{\sqrt{625}}{\sqrt{16}}=\boxed{\dfrac{25}{4}}[/tex]
b)
[tex]\sqrt{3^{2}+\dfrac{3^{4}}{2^{4}}}=\sqrt{\dfrac{16\cdot9+81}{16}}=\sqrt\dfrac{144+81}{16}}=\dfrac{\sqrt{225}}{\sqrt{16}}=\boxed{\dfrac{15}{4}}[/tex]
c)
[tex]\sqrt{7^{2}+\dfrac{7^{2}\cdot4^{2}}{9}}=\sqrt{\dfrac{49\cdot9+49\cdot16}{16}}=\sqrt{\dfrac{1225}{9}}=\boxed{\dfrac{35}{3} }[/tex]
d)
[tex]\sqrt{4^{2}+\dfrac{4^{2}\cdot5^{2}}{12^{2}}}=\sqrt{\dfrac{16\cdot144+16\cdot25}{144}}=\sqrt{\dfrac{2304+400}{144}}=\sqrt{\dfrac{2704}{144}}=\boxed{\dfrac{13}{3} }[/tex]
e)
[tex]\sqrt{\dfrac{3^{2}\cdot13^{2}}{5^{2}}-3^{2}}=\sqrt{\dfrac{9\cdot169-25\cdot9}{25}}=\sqrt{\dfrac{1521-225}{25}}=\sqrt\dfrac{1296}{25}}= \boxed{\dfrac{36}{5} }[/tex]
f)
[tex]\sqrt{\dfrac{5^{2}\cdot8^{2}}{3^{2}}+5^{2}\cdot 2^{2}}=\sqrt{\dfrac{25\cdot64+25\cdot4\cdot9}{9}} =\sqrt{\dfrac{1600+900}{9}}=\sqrt{\dfrac{2500}{9}}=\boxed{\dfrac{50}{3} }[/tex]
g)
[tex]\sqrt{\dfrac{7^{2}\cdot8^{2}}{15^{2}}+7^{2} }=\sqrt{\dfrac{49\cdot64+49\cdot225}{225}}=\sqrt{\dfrac{3136+11025}{225}}=\sqrt{\dfrac{14161}{225}}=\boxed{\dfrac{119}{15} }[/tex]
