N. 257
IN 3 PASSI
(1) Trasforma tutte le potenze nella stessa base $\frac{2}{3}$.
(2) Applica la terza proprietà delle potenze:
$\left(a^n\right)^m=a^{n \cdot m}$, determinando anche il segno di ciascuna potenza.
(3) Applica la seconda proprietà delle potenze.
124
punti
Livello 1
16 = $4^2$ = $(2^2)^2 = 2^{2\cdot 2} = 2^4$ , $81 = 9^2 = 3^4$. Quindi il tutto diventa
$ \left [ \dfrac{2^{12}}{3^{12}} \div (-1)^2 \dfrac{2^4}{3^4} \right ]^2 \div \left (\dfrac{2}{3} \right )^{14}$
$\left [ \left (\dfrac{2}{3}\right )^{12} \div \left (\dfrac{2}{3} \right )^4 \right ]^2 \div \left (\dfrac{2}{3} \right )^{14}$
$\left [ \left ( \dfrac{2}{3}\right )^{12-4}\right ]^2 \div \left (\dfrac{2}{3}\right )^{14}$
$ \left [ \left ( \dfrac{2}{3}\right )^{8}\right ]^2 \div \left (\dfrac{2}{3} \right )^{14}$
$ \left (\dfrac{2}{3}\right )^{16} \div \left (\dfrac{2}{3}\right )^{14}$
$ \left (\dfrac{2}{3}\right )^{16-14} =\left ( \dfrac{2}{3} \right )^{2} = 4/9 $
3729
punti
Livello 5
((16/81)^3/(- 4/9)^2)^2/(- (2/3)^7)^2=
=((2^4/3^4)^3/(- 2^2/3^2)^2)^2/(- 2^7/3^7)^2=
=(2^12/3^12/(2^4/3^4))^2/(2^14/3^14)=
=(2^8/3^8)^2/(2^14/3^14)=
=2^16/3^16/(2^14/3^14)=
=2^2/3^2= 4/9
115473
punti
Genius III
((- 7/3)^2·(49/9)^2)^2/(- 7/3)^13=
=(7^2/3^2·(7^2/3^2)^2)^2/(- 7^13/3^13)=
=(7^2/3^2·(7^4/3^4))^2/(- 7^13/3^13)=
=(7^6/3^6)^2/(- 7^13/3^13)=
=7^12/3^12/(- 7^13/3^13)=
=7^12/3^12·(- 3^13/7^13) = - 3/7
115473
punti
Genius III
