[tex]a)~~( 5^{\sqrt{3} } )^{\sqrt{3} } =5^{\sqrt{3}\cdot \sqrt{3} } =5^{3 } =125\\\\b)~~( 3^{\sqrt{2} } )^{2\sqrt{2} } =3^{\sqrt{2}\cdot 2\sqrt{2} }=3^{4 }=81\\\\c)~~( 5^{\sqrt{3} -1} )^{\sqrt{3} +1} =5^{(\sqrt{3} -1)\cdot ( \sqrt{3} +1 ) } =5^{(\sqrt{3} )^{2} -1^{2} } =5^{3-1} =5^{2} =25\\\\d)~~( 2^{\sqrt{7} -\sqrt{2} } )^{\sqrt{7} +\sqrt{2} } =2^{(\sqrt{7} -\sqrt{2} )\cdot ( \sqrt{7} +\sqrt{2} ) } =2^{(\sqrt{7} )^{2} -( \sqrt{2} )^{2} } =2^{7-2} =2^{5} =32\\\\[/tex]
[tex]e)~~7^{\sqrt{2} } \cdot 49^{-\frac{\sqrt{2} }{2} } =7^{\sqrt{2} } \cdot ( 7^{2} )^{-\frac{\sqrt{2} }{2} } =7^{\sqrt{2} } \cdot 7^{2\cdot (-\frac{\sqrt{2} }{2} ) } =7^{\sqrt{2} } \cdot 7^{-\sqrt{2} } =7^{\sqrt{2} \cdot ( - \sqrt{2} )} =7^{-2} =( \dfrac{1}{7} ) ^{2} = \dfrac{1}{49}[/tex]
[tex]f)~~9^{\sqrt{5} } \cdot 3^{1-2\sqrt{5} } =9^{\sqrt{5} } \cdot \dfrac{3^{1} }{3^{2\sqrt{5} } } =9^{\sqrt{5} } \cdot \dfrac{3 }{(3^{2} )^{\sqrt{5} } }=9^{\sqrt{5} } \cdot \dfrac{3 }{9^{\sqrt{5} } }=3[/tex]
Korzystam ze wzorów:
[tex]( x^{n} )^{m} =x^{n\cdot m} \\\\x^{n} \div x^{m} = \dfrac{x^{n} }{x^{m} } =x^{n-m} \\\\(x-y) \cdot (x+y) =x^{2} -y^{2} \\\\x^{-n} =(\dfrac{1}{x} )^{n}[/tex]
