So sánh 2^225 và 3^150

\(\text{Câu 1 :}\)

\(\text{a)}\) \(2^{225}\) \(\text{và}\) \(3^{150}\)

\(\text{Ta có :}\)\(2^{225}=\left(2^3\right)^{75}=8^{75}\)

\(3^{150}=\left(3^2\right)^{75}=9^{75}\)

\(\text{Mà}\) \(8^{75}< 9^{75}\)

\(\Rightarrow2^{225}< 3^{150}\)

\(\text{Vậy}\) \(2^{225}< 3^{150}\)

\(\text{b)}\) \(2^{91}\) \(\text{và}\) \(5^{35}\)

\(\text{Ta có :}\) \(2^{91}=\left(2^{13}\right)^7=8192^7\)

\(5^{35}=\left(5^5\right)^7=3125^7\)

\(\text{Mà}\) \(8192^7>3125^7\) \(\)

\(\Rightarrow2^{91}>5^{35}\)

\(\text{Vậy}\) \(2^{91}>5^{35}\)

\(\text{c)}\) \(3^{4000}\) \(\text{và}\) \(9^{2000}\)

\(\text{Ta có :}\) \(9^{2000}=\left(3^2\right)^{2000}=3^{4000}\)

\(Mà\) \(3^{4000}=3^{4000}\)

\(\text{Vậy}\) \(3^{4000}=9^{2000}\)

\(\text{d)}\) \(2^{332}\) \(\text{và}\) \(3^{223}\)

\(\text{Ta có : }\) \(2^{332}< 2^{333}\)

\(3^{223}>3^{222}\) \(\left(1\right)\)

\(\text{Ta lại có : }\) \(2^{333}=\left(2^3\right)^{111}=8^{111}\)

\(3^{222}=\left(3^2\right)^{111}=9^{111}\)

\(\text{Mà}\) \(8^{111}< 9^{111}\)

\(\Rightarrow2^{333}< 3^{222}\) \(\left(2\right)\)

\(\text{Từ}\) \(\left(1\right)\) \(\text{và}\) \(\left(2\right)\) \(\text{suy ra : }\)

\(2^{322}< 2^{333}< 3^{222}< 3^{223}\)

\(\Rightarrow2^{332}< 3^{223}\)

\(\text{Vậy}\) \(2^{332}< 3^{223}\)

\(\text{Câu 2 :}\)

\(\text{a)}\) \(\dfrac{120^3}{40^3}=\left(\dfrac{120}{40}\right)^3=3^3=27\)

\(\text{b)}\) \(\dfrac{45^{10}\cdot5^{20}}{75^{15}}=\dfrac{\left(9\cdot5\right)^{10}\cdot5^{20}}{\left(3\cdot25\right)^{15}}\)\(=\dfrac{9^{10}\cdot5^{10}\cdot5^{20}}{3^{15}\cdot25^{15}}\)\(=\dfrac{\left(3^2\right)^{10}\cdot5^{30}}{3^{15}\cdot\left(5^2\right)^{15}}\)\(=\dfrac{3^{20}\cdot5^{30}}{3^{15}\cdot5^{30}}\)\(=\dfrac{3^{20}}{3^{15}}\)\(=3^4=81\)

\(\text{c)}\) \(\dfrac{390^4}{130^4}=\left(\dfrac{390}{130}\right)^4=3^4=81\)