[tex]d) \ 3^{2}\cdot5^{6}+4^{2}\cdot5^{6} = 5^{6}(3^{2}+4^{2}) = 5^{6}(9+16) = 5^{6}\cdot25 = 5^{6}\cdot5^{2} = 5^{6+2} = 5^{8}\\\\\\e) \ 5^{1}\cdot5^{6}+5^{2}\cdot5^{5}+5^{3}\cdot5^{4}+5^{4}\cdot5^{3}+5^{5}\cdot5^{2} =5^{1+6}+5^{2+5}+5^{3+4}+5^{4+3}+5^{5+2} =\\\\=5^{7}+5^{7}+5^{7}+5^{7}+5^{7} =5\cdot5^{7} = 5^{1+7} = 5^{8}[/tex]
[tex]f) \ \frac{5^{2}\cdot125}{5^{0}+24}=\frac{5^{2}\cdot5^{3}}{1+24} = \frac{5^{2+3}}{25} = \frac{5^{5}}{5^{2}} = 5^{5-2} = 5^{3}[/tex]
Skorzystano z własności potęgowania:
[tex]a^{m}\cdot a^{n} = a^{m+n}\\\\a^{m}:a^{n} = a^{m-n}\\\\(a^{m})^{n} = a^{m\cdot n}\\\\a^{0} = 1[/tex]
