Question

Determinati numarul natural n pentru care au loc egalitatile:
a)3 la puterea 2*n-3=27 la puterea 7:9 la puterea 2=
b)5 la puterea 3*n+1=25 la puterea11=
c)7 la puterea n+1:7 la puterea 4=49 la puterea 2 totul la puterea 3=
d)10 la puterea 2*n+1:5 la puterea 2*n+1=8 la puterea 9=
Va rog repede!!

Answer 1

$a).3^{2n-3}=27^7:9^2 \\ 3^{2n-3}=(3^3)^7:(3^2)^2 \\ 3^{2n-3}=3^2^1:3^4 \\ 3^{2n-3}=3^{21-4} \\ 3^{2n-3}=3^1^7 \\ 2n-3=17 \\ 2n=17+3 \\ 2n=20 \\ n= 20:2 \\ n=10$
$b).5^{3n+1}=25^1^1 \\ 5^{3n+1}=(5^2)^1^1 \\ 5^{3n+1}=5^2^2 \\ 3n+1=22 \\ 3n=22-1 \\ 3n=21 \\ n= 21:3 \\ n=7$
$c).7^{n+1}:7^4= 49^{ 2^{3} } \\ 7^{n+1}:7^4=49^8 \\ 7^{n+1-4}=(7^2)^8 \\ 7^{n-3}=7^1^6 \\ n-3=16 \\ n=16+3 \\ n=19$
$d).10^{2n+1}:5^{2n+1}=8^9 \\ 2^{2n+1} \cdot5^{2n+1}:5^{2n+1}=(2^3)^9 \\ 2^{2n+1}=2^2^7 \\ 2n+1=27 \\ 2n=27-1 \\ 2n=26 \\ n=26:2 \\ n=13$