Question

tìm số nguyên n lớn nhất sao cho \(n^{150}\)<\(5^{225}\)

Answer 1

\(n^{150}< 5^{225}\)

\(\Leftrightarrow\left(n^{30}\right)^5< \left(5^{45}\right)^5\)

\(\Leftrightarrow n^{30}< 5^{45}\)

\(\Leftrightarrow\left(n^{10}\right)^3< \left(5^{15}\right)^3\)

\(\Rightarrow n^{10}< 5^{15}\)

\(\Leftrightarrow\left(n^2\right)^5< \left(5^3\right)^5\)

\(\Rightarrow n^2< 125\)

\(MAX_n\Rightarrow MAX_{n^2}\)

\(\Rightarrow n^2=121\Rightarrow n=11\)