zadanie 1
a)
\frac{5^4*5^{-2}}{5^3}=5^{4-3}*5^{-2}=5^{1+(-2)}=5^{-1}
b)
25^{\frac{2}{3}}+32^{\frac{2}{5}}=\sqrt[3]{25^2}+(2^5)^{\frac{2}{5}}=\sqrt[3]{(5^2)^2}+2^2=
\sqrt[3]{5^3*5}+4=5\sqrt[3]5+4
c)
5^{-\frac{2}{7}}:25^{-\frac{1}{7}}=5^{-\frac{2}{7}}:(5^2)^{-\frac{1}{7}}=5^{-\frac{2}{7}}:5^{-\frac{2}{7}}=1
log_{27}3\sqrt3=x
27^x=3\sqrt3
(3^3)^x=3*3^{\frac{1}{2}}
3^{3x}=3^{1+\frac{1}{2}}
3^{3x}=3^{\frac{3}{2}}
3x=\frac{3}{2} |*2
6x=3
x=\frac{3}{6}
x=\frac{1}{2}
log_ab=c , a^c=b
Zadanie 2
a)
16\sqrt[3]{16}=16*2\sqrt[3]2=32\sqrt[3]2=2^5*2^{\frac{1}{3}}=2^{\frac{15}{3}+\frac{1}{3}}=2^{\frac{16}{3}}
b)
\sqrt[4]{3\sqrt3}=\sqrt[4]{3*3^{\frac{1}{2}}}=\sqrt[4]{3^{\frac{3}{2}}}=
(3^{\frac{3}{2}})^{\frac{1}{4}}=3^{\frac{3}{8}}