Question

trong hai biểu thức:

\(\dfrac{3^6.21^{12}}{175^9.7^3}\)và \(\dfrac{3^{10}.6^7.4}{10^9.5^8}\)

biểu thức nào có giá trị lớn hơn?

Answer 1

Đặt\(A=\dfrac{3^6.21^{12}}{175^9.7^3}=\dfrac{3^6.3^{12}.7^{12}}{175^9.7^3}=\dfrac{3^{18}.7^{12}}{\left(5^2\right)^9.7^9.7^3}=\dfrac{3^{18}.7^{12}}{5^{18}.7^{12}}=\dfrac{3^{18}}{5^{18}}=\left(\dfrac{3}{5}\right)^{18}\)

Đặt \(B=\dfrac{3^{10}.6^7.4}{10^9.5^8}=\dfrac{3^{10}.3^7.2^7.2^2}{2^9.5^9.5^8}=\dfrac{3^{17}.2^9}{2^9.5^{17}}=\dfrac{3^{17}}{5^{17}}=\left(\dfrac{3}{5}\right)^{17}\)

Mà \(\left(\dfrac{3}{5}\right)^{18}>\left(\dfrac{3}{5}\right)^{17}\Leftrightarrow A>B\)

\(\Rightarrow\dfrac{3^6.21^{12}}{175^9.7^3}>\dfrac{3^{10}.6^7.4}{10^9.5^8}\)