Question

Chứng minh rằng 2^1995<5^863

Answer 1

\(2^{1995}=2^5\cdot2^{1990}=32\cdot2^{1990}\)

\(5^{863}=5^{860}\cdot5^3=5^{860}\cdot125\)

Ta có: \(2^{10}\cdot3=1024\cdot3=3072;5^5=3125\)

Do đó: \(2^{10}\cdot3<5^5\)

=>\(\left(2^{10}\cdot3\right)^{172}<\left(5^5\right)^{172}\)

=>\(2^{1720}\cdot3^{172}<5^{860}\) (1)

Ta có: \(3^7=2187;2^{11}=2048\)

=>\(3^7>2^{11}\)

\(3^{172}=\left(3^7\right)^{24}\cdot3^4>\left(2^{11}\right)^{24}\cdot3^4=2^{270}\)

=>\(2^{1720}\cdot2^{270}<2^{1720}\cdot3^{172}\) (2)

Từ (1),(2) suy ra \(2^{1720}\cdot2^{270}<5^{860}\)

=>\(2^{1995}<5^{863}\)