Zadanie 1.
a.
16^4=16^{2+2}=16^2*16^2=65536
b.
(\frac{1}{3})^4*3^4*6^4=\frac{1^4}{3^4}*3^4*6^{2+2}=1*6^2*6^2=36*36=
c.
(0,2)^{-2} * 2^{-2} * (\frac{1}{4})^{-2}=(\frac{10}{2})^2*(\frac{1}{2})^2*(\frac{4}{1})^2=\frac{10^2}{2^2}*\frac{1^2}{2^2}*4^2=\frac{10^2}{2^2}*\frac{1}{2^2}*2^2*2^2=100
d.
16^3 *(\frac{1}{8})^3 * 3^3 * (\frac{1}{9})^3=(2*8)^3*\frac{1^3}{8^3}*3^3*\frac{1^3}{(3*3)^3}=2^3*8^3*\frac{1}{8^3}*3^3*\frac{1}{3^3*3^3}=2^3*1*\frac{1}{3^3}=\frac{8}{9}
e.
27^2 * (\frac{1}{9})^2 * (\frac{1}{3})^2=(9*3)^2*\frac{1^2}{9^2}*\frac{1^2}{3^2}=9^2*3^2*\frac{1}{9^2}*\frac{1}{3^2}=1
Zadanie 2.
a.
6^2 * 3^2 * 2^2=(6*3*2)^2=36^2=(6^2)^2=6^{2*2}=6^4
b.
(7,2)^3 * 3^3 * (\frac{1}{4})^3=(\frac{72}{10}*3)^3=(\frac{24}{10})^3=(2,4)^3
c.
(\frac{11}{16})^{-2} * (\frac{16}{11})^-2 * (0,2)^{-2}=\frac{16^{2}}{11^{2}}*\frac{16^2}{11^2}*\frac{10^2}{2^2}=
d.
(0,2)^4 * 5^4=(0,2*5)^4=1^4=1
e.
(1,5)^{-3} * (0,4)^{-3}=(1,5*0,4)^{-3}=(\frac{6}{10})^{-3}=\frac{10^3}{6^3}=(\frac{10}{6})^3=(\frac{5}{3})^3=(1\frac{2}{3})^3
f.
2^4 * 5^4=(2*4)^4=8^4
Zadanie 3.
a.
(\frac{1}{4}^{10})* (\frac{4}{3})^{10} * 36^{10}=\frac{1^4}{4^{10}}*\frac{4^{10}}{3^{10}}*36^{10}=\frac{36^{10}}{3^{10}}=\frac{3^{10}*12^{10}}{3^{10}}=12^{10}
b.
(0,55)^2 * (\frac{1}{11})^2=\frac{55^2}{100^2}*\frac{1^2}{11^2}=\frac{5^2*11^2}{10^2}*\frac{1}{11^2}=\frac{5^2}{10^2}=\frac{5^2}{2^2*5^2}=\frac{1}{2^2}=(\frac{1}{2})^2=2^{-2}
c.
2^{30}*3^{20}*(\frac{1}{2})^{20}*(\frac{2}{3})^{20}=2^{30}*(3*\frac{1}{2}*\frac{2}{3})^{20}=2^{30}*1^{20}=2^{30}
d.
(1 \frac{1}{3})^{-10} * (0,75)^{-10}=(\frac{4}{3}*\frac{3}{4})^{-10}=1^{-10}=1
e.
(17 \frac{1}{2})^3 * (0,4)^3 * (\frac{1}{7})^3=(\frac{35}{2}*\frac{4}{10}*\frac{1}{7})^3=(\frac{5*2}{10})^3=(\frac{10}{10})^3=1^3=1