Đặt \(A=\frac{2^{19}\cdot27^3+15\cdot4^9\cdot9^4}{6^9\cdot2^{10}+12^{10}}\)

\(A=\frac{2^{19}\cdot\left(3^3\right)^3+15\cdot\left(2^2\right)^9\cdot\left(3^2\right)^4}{6^9\cdot2^9\cdot2+12^{10}}\)

\(A=\frac{2^{19}\cdot3^9+15\cdot2^{18}\cdot3^8}{12^9\cdot2+12^9\cdot12}=\frac{\left(2^{18}\cdot3^8\right)\cdot6+\left(2^{18}\cdot3^8\right)\cdot15}{12^9\cdot\left(2+12\right)}\)

\(A=\frac{\left(2^{18}\cdot3^8\right)\cdot\left(6+15\right)}{12^9\cdot14}=\frac{2^{18}\cdot3^8\cdot21}{12^9\cdot14}=\frac{2^{18}\cdot3^8\cdot7\cdot3}{2^{18}\cdot3^9\cdot7\cdot2}=\frac{3^8\cdot3}{3^8\cdot3\cdot2}\)

\(A=\frac{1}{2}\)

Đặt \(B=\frac{4}{35}+\frac{4}{63}+\frac{4}{99}+\frac{4}{143}+\frac{4}{195}=\frac{4}{5\cdot7}+\frac{4}{7\cdot9}+\frac{4}{9\cdot11}+\frac{4}{11\cdot13}+\frac{4}{13\cdot15}\)

\(B=\frac{1}{2}\left(\frac{4}{5}-\frac{4}{7}+\frac{4}{7}-\frac{4}{9}+...+\frac{4}{13}-\frac{4}{15}\right)\)

\(B=\frac{1}{2}\left(\frac{4}{5}-\frac{4}{15}\right)\)mà \(\frac{4}{5}-\frac{4}{15}< 1\Leftrightarrow\frac{1}{2}\left(\frac{4}{5}-\frac{4}{15}\right)< \frac{1}{2}\Leftrightarrow B< A\)

may mà m đăng để t đỡ phải viết (mỏi tay) Nguyen Ngoc Linh

\(A=\dfrac{2^{19}.27^3+15.4^9.9^4}{6^9.2^{10}+2^{12}}\)

\(=\dfrac{2^{19}.3^9+5.2^{18}.3^9}{2^9.3^{19}+2^{12}}=\dfrac{2^{10}+5.2^8}{3^{10}+2^3}=\dfrac{2^7+5.2^5}{3^{10}}\)

\(B=\dfrac{4}{35}+\dfrac{4}{63}+\dfrac{4}{99}+\dfrac{4}{143}+\dfrac{4}{195}\)

\(=2\left(\dfrac{2}{5.7}+\dfrac{2}{7.9}+\dfrac{2}{9.11}+\dfrac{2}{11.13}+\dfrac{2}{13.15}\right)\)

\(=2.\left(\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{15}\right)\)

\(=2\left(\dfrac{1}{5}-\dfrac{1}{15}\right)=2\left(\dfrac{2}{15}\right)\)=\(\dfrac{4}{15}\)

\(a,\left(-2\right)^{300}=2^{300}=\left(2^3\right)^{100}=8^{100}\)

\(\left(-3\right)^{200}=3^{200}=\left(3^2\right)^{100}=9^{100}\)

\(\Leftrightarrow\left(-3\right)^{200}>\left(-2\right)^{300}\)

\(b,A=2^{19}.27^3+15.4^9.9^4\)

=> A=2¹⁸.3³⁹ B=\(\dfrac{4}{15}\)

vì A>1>B

=> B<A