Question

Tìm m,n \(\in\)N* thỏa mãn : 2^m + 2ⁿ= 2^m+n

Answer 1

Ta có:

\(2^m+2^n=2^{m+n}\\ \Leftrightarrow2^{m+n}-2^m-2^n=0\\ \Leftrightarrow2^m.2^n-2^m-2^n=0\\ \Leftrightarrow2^m\left(2^n-1\right)-2^n=0\\ \Leftrightarrow2^m\left(2^n-1\right)-\left(2^n-1\right)-1=0\\ \Leftrightarrow\left(2^n-1\right)\left(2^m-1\right)=1\\ \Leftrightarrow\left\{{}\begin{matrix}2^m-1=1\\2^n-1=1\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}2^m=2\\2^n=2\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}m=1\\n=1\end{matrix}\right.\)

(T/m \(m,n\in N\)*)

Vậy m = n = 1.