Question

determinati numerele naturale n stiind ca:
a) 3ⁿ=7ⁿ
b) 2ⁿ=4³
c) 3²ⁿ<9³
d)7ⁿ=1

Answer 1

   
[tex]\displaystyle \\ a)~~\text{Daca doua puteri egale cu baze diferite si exponenti egali, atunci} \\ \text{exponentii sunt egali cu zero.} \\ 3^n=7^n ~~\Longrightarrow~~\boxed{n=0} \\ \\ b) \\ 2^n=4^3\\ 2^n =(2^2)^3 \\ 2^n =2^{2\times3} \\ 2^n =2^6 ~~\Longrightarrow~~\boxed{n=6} [/tex]


[tex]c) \\ 3^{2n} \ \textless \ 9^3 \\ 3^{2n} \ \textless \ (3^2)^3 \\ 3^{2n} \ \textless \ 3^{2\times 3} \\ 3^{2n} \ \textless \ 3^6 \\ 2n \ \textless \ 6 ~~| : 2 \\ n \ \textless \ 3 ~~\Longrightarrow~~\boxed{n \in \{0;~1;~2\} } \\ \\ d) 7^n=1 ~~\Longrightarrow~~\boxed{n=0}[/tex]