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$\sqrt{3 \sqrt{5} }*\sqrt[4]{1125}:( \sqrt[3]{5}* \sqrt{5 \sqrt[3]{5} })=\\\\=3^{1/2}*(5^{1/2})^{1/2}*(5^3*3^2)^{1/4}:(5^{1/3}*5^{1/2}*(5^{1/3})^{1/2})=\\\\=3^{1/2}*5^{1/4}*5^{3/4}*3^{1/2}:(5^{1/3}*5^{1/2}*5^{1/6})=3*5:5=3$

$\sqrt{5 \sqrt{5} }: \sqrt{ \sqrt[3]{5 \sqrt{5}} * \sqrt[4]{5 \sqrt[3]{5}}}=\\\\=5^{1/2}*(5^{1/2})^{1/2}:((5^{1/3})^{1/2}*((5^{1/2})^{1/3})^{1/2}*(5^{1/4})^{1/2}*((5^{1/3})^{1/4})^{1/2})\\\\=5^{1/2}*5^{1/4}:(5^{1/6}*5^{1/12}*5^{1/8}*5^{1/24})=\\\\=5^{3/4}:5^{(4+2+3+1)/24}=5^{3/4}:5^{10/24}=5^{3/4}:5^{5/12}=\\\\=5^{9/12 -5/12}=5^{4/12}=5^{1/3}= \sqrt[3]{5}$