L \ - \ lewa strona równania
I. P
\frac{\frac{1}{2} \cdot 5^{\frac{1}{3}} \cdot 4^{\frac{1}{5}}}{(675^{\frac{1}{12}} \cdot 3^{\frac{1}{4}} \cdot 2^{0,3})^{-2}}=3 \cdot 5^{\frac{2}{3}}
L= \frac{2^{-1} \cdot 5^{\frac{1}{3}} \cdot (2^2)^{\frac{1}{5}}}{[(3^3 \cdot 5^2)^{\frac{1}{12}} \cdot 3^{\frac{1}{4}} \cdot 2^{0,3}]^{-2}}\\ = \frac{2^{-1+\frac{2}{5}} \cdot 5^{\frac{1}{3}}}{(3^{\frac{1}{4}})^{-2} \cdot (3^{\frac{1}{4}})^{-2}\cdot (5^{\frac{1}{6}})^{-2}\cdot 2^{-0,6}}\\ =\frac{2^{-0,6} \cdot 5^{\frac{1}{3}}}{3^{-\frac{1}{2}-\frac{1}{2}} \cdot 5^{-\frac{1}{3}} \cdot 2^{-0,6}} \\ = \frac{5^{\frac{1}{3}+\frac{1}{3}}}{3^{-1}} \\ = 3 \cdot 5^{\frac{2}{3}}
3 \cdot 5^{\frac{2}{3}}=3 \cdot 5^{\frac{2}{3}}
Dodatkowe obliczenia
Rozkładam 675 na czynniki pierwsze
675|3
225|3
75 | 3
25 | 5
5 | 5
1
675 = 3^3 * 5^2
II. F
\frac{\sqrt2 \cdot 4^{\frac{1}{3}} \cdot 8^{-\frac{1}{3}}}{16^{\frac{1}{3}} \cdot 2^{\frac{1}{4}}}=2^{\frac{17}{12}} \\ L= \frac{2^{\frac{1}{2}} \cdot (2^2)^{\frac{1}{3}} \cdot (2^3)^{-\frac{1}{3}}}{(2^4)^{\frac{1}{3}} \cdot 2^{\frac{1}{4}}}\\ =\frac{2^{\frac{1}{2}} \cdot 2^{\frac{2}{3}} \cdot 2^{-1}}{2^{\frac{4}{3}} \cdot 2^{\frac{1}{4}}} =\frac{2^{\frac{6+8-12}{12}}}{2^{\frac{16+3}{12}}} \\ =\frac{2^{\frac{2}{12}}}{2^{\frac{19}{12}}} = 2^{\frac{2}{12}-\frac{19}{12}}=2^{-\frac{17}{12}}
2^{-\frac{17}{12}} \ne 2^{\frac{17}{12}}
III. P
3\sqrt{27} \cdot 9^{-1,5} \cdot (\frac{1}{3})^{\frac{3}{4}} \cdot (\frac{1}{81})^{-2}= 3^{\frac{27}{4}}\\ L=3 \cdot (3^3)^{\frac{1}{2}} \cdot (3^{2})^{-1,5} \cdot (3^{-1})^{\frac{3}{4}} \cdot (3^{-4})^{-2}\\ =3^1 \cdot 3^{\frac{3}{2}} \cdot 3^{-3} \cdot 3^{-\frac{3}{4}} \cdot 3^{8} \\ =3^{\frac{4+6-12-3+32}{4}}
=3^{\frac{27}{4}} \qquad
3^{\frac{27}{4}} =3^{\frac{27}{4}}