[tex]a)\\\\\frac{5^{13}*3+2*5^{13}} {(5^{11}:5^{4}) *25} = \frac{5^{13} *(3+2) } { 5^{11-7} * 5^2}= \frac{5^{13} *5 } { 5^{7} * 5^2}= \frac{5^{13+1} } { 5^{7+2} } = \frac{5^{14} } { 5^{9} } =5^{14-9}=5^{5}\\\\stosujemy\ wzory:\\\\a^n*a^m =a^{n+m}\\\\ a^n:a^m =a^{n-m}[/tex]
[tex]b)\\\\ 2^{\frac{1}{2}}*50 ^{\frac{1}{2}}-2 ^{\frac{1}{5}}*16^{\frac{1}{5}} =(2*50)^{\frac{1}{2}} -(2*16)^{\frac{1}{5}}=100^{\frac{1}{2}}-32^{\frac{1}{5}}= \\\\=\sqrt{100}-\sqrt[5]{32}=10-2=8 \\\\ stosujemy\ wzor:\\\\ a^n*b^n=(a*b)^n[/tex]
[tex]c)\\\\ \sqrt[3]{375}-\sqrt[3]{-192}-\sqrt[3]{-81}=\sqrt[3]{125*3}+\sqrt[3]{ 64*3}+\sqrt[3]{ 27*3}=\\\\=\sqrt[3]{125 }*\sqrt[3]{3}+\sqrt[3]{ 64 }*\sqrt[3]{3}+\sqrt[3]{ 27 }*\sqrt[3]{3}=\sqrt[3]{ 5^3 }*\sqrt[3]{3}+\sqrt[3]{4^3 }*\sqrt[3]{3}+\sqrt[3]{ 3^3}*\sqrt[3]{3}=\\\\=5 \sqrt[3]{3}+4\sqrt[3]{3}+3\sqrt[3]{3}=12\sqrt[3]{3}[/tex]
[tex]d)\\\\ 5^{2+\sqrt{3}} :5^{ 3+ \sqrt{3} }=5^{2+\sqrt{3}-(3+\sqrt{3})}=5^{2+\sqrt{3}-3-\sqrt{3}}=5^{-1}=\frac{1}{5^1}=\frac{1}{5}\\\\stosujemy\ wzory:\\\\a^n:a^m=a^{n-m}\\\\a^{-n}=\frac{1}{a^n}[/tex]
