Bài 1 :
a) Ta có :
\(2^{225}=\left(2^3\right)^{75}=8^{75}\)
\(3^{150}=\left(3^2\right)^{75}=9^{75}\)
Vì \(8^{75}< 9^{75}\Leftrightarrow2^{225}< 3^{150}\)
b) Ta có :
\(2^{91}=\left(2^{13}\right)^7=8192^7\)
\(5^{35}=\left(5^5\right)^7=3125^7\)
Vì \(8192^7>3125^7\Leftrightarrow2^{91}>5^{35}\)
c)Ta có :
\(3^{4000}=\left(3^4\right)^{1000}=81^{1000}\)
\(9^{2000}=\left(9^2\right)^{1000}=81^{1000}\)
Vì \(81^{1000}=81^{1000}\Leftrightarrow3^{4000}=9^{2000}\)
d) Ta có :
\(2^{332}< 2^{333}=\left(2^3\right)^{111}=8^{111}\)
\(3^{223}< 3^{222}=\left(3^2\right)^{111}=9^{111}\)
Mà \(8^{111}< 9^{111}\Leftrightarrow2^{332}< 3^{223}\)
Bài 2 :
a) \(\dfrac{120^3}{40^3}=\left(\dfrac{120}{4}\right)^3=3^3=27\)
b) \(\dfrac{390^4}{130^4}=\left(\dfrac{390}{130}\right)^4=3^4=81\)
c) \(\dfrac{45^{10}.5^{20}}{75^{15}}=\dfrac{\left(3^2.5\right)^{10}.5^{20}}{\left(3.5^2\right)^{15}}=\dfrac{3^{20}.5^{10}.5^{20}}{3^{15}.5^{30}}=3^5=243\)
Bài 1:
a.Ta có :
\(2^{225}=\left(2^3\right)^{75}=8^{75}\)
\(3^{150}=\left(3^2\right)^{75}=9^{75}\)
Vì \(8^{75}< 9^{75}\) nên \(2^{225}< 3^{150}\)
b. Ta có :
\(2^{91}=\left(2^{13}\right)^7=8192^7\)
\(5^{35}=\left(5^5\right)^7=3125^7\)
Vì \(8192^7>3125^7\) nên \(2^{91}>5^{35}\)
c. Ta có :
\(3^{4000}=\left(3^2\right)^{2000}=9^{2000}\)
Vì \(9^{2000}=9^{2000}\) nên \(3^{4000}=9^{2000}\)
Bài 2:
a. \(\dfrac{120^3}{30^3}=\dfrac{\left(30.4\right)^3}{30^3}=\dfrac{30^3.4^3}{30^3}=4^3=64\)
b. \(\dfrac{45^{10}.5^{20}}{75^{15}}=\dfrac{\left(5.3^2\right)^{10}.5^{20}}{\left(3.5^2\right)^{15}}=\dfrac{5^{10}.3^{20}.5^{20}}{3^{15}.5^{30}}=\dfrac{5^{30}.3^{20}}{3^{15}.5^{30}}=3^5=243\)
c. \(\dfrac{390^4}{130^4}=\dfrac{\left(130.3\right)^4}{130^4}=\dfrac{130^4.3^4}{130^4}=3^4=81\)
\(\text{Câu 1 :}\)
\(\text{a)}\) \(2^{225}\) \(\text{và}\) \(3^{150}\)
\(\text{Ta có :}\)\(2^{225}=\left(2^3\right)^{75}=8^{75}\)
\(3^{150}=\left(3^2\right)^{75}=9^{75}\)
\(\text{Mà}\) \(8^{75}< 9^{75}\)
\(\Rightarrow2^{225}< 3^{150}\)
\(\text{Vậy}\) \(2^{225}< 3^{150}\)
\(\text{b)}\) \(2^{91}\) \(\text{và}\) \(5^{35}\)
\(\text{Ta có :}\) \(2^{91}=\left(2^{13}\right)^7=8192^7\)
\(5^{35}=\left(5^5\right)^7=3125^7\)
\(\text{Mà}\) \(8192^7>3125^7\) \(\)
\(\Rightarrow2^{91}>5^{35}\)
\(\text{Vậy}\) \(2^{91}>5^{35}\)
\(\text{c)}\) \(3^{4000}\) \(\text{và}\) \(9^{2000}\)
\(\text{Ta có :}\) \(9^{2000}=\left(3^2\right)^{2000}=3^{4000}\)
\(Mà\) \(3^{4000}=3^{4000}\)
\(\text{Vậy}\) \(3^{4000}=9^{2000}\)
\(\text{d)}\) \(2^{332}\) \(\text{và}\) \(3^{223}\)
\(\text{Ta có : }\) \(2^{332}< 2^{333}\)
\(3^{223}>3^{222}\) \(\left(1\right)\)
\(\text{Ta lại có : }\) \(2^{333}=\left(2^3\right)^{111}=8^{111}\)
\(3^{222}=\left(3^2\right)^{111}=9^{111}\)
\(\text{Mà}\) \(8^{111}< 9^{111}\)
\(\Rightarrow2^{333}< 3^{222}\) \(\left(2\right)\)
\(\text{Từ}\) \(\left(1\right)\) \(\text{và}\) \(\left(2\right)\) \(\text{suy ra : }\)
\(2^{322}< 2^{333}< 3^{222}< 3^{223}\)
\(\Rightarrow2^{332}< 3^{223}\)
\(\text{Vậy}\) \(2^{332}< 3^{223}\)
\(\text{Câu 2 :}\)
\(\text{a)}\) \(\dfrac{120^3}{40^3}=\left(\dfrac{120}{40}\right)^3=3^3=27\)
\(\text{b)}\) \(\dfrac{45^{10}\cdot5^{20}}{75^{15}}=\dfrac{\left(9\cdot5\right)^{10}\cdot5^{20}}{\left(3\cdot25\right)^{15}}\)\(=\dfrac{9^{10}\cdot5^{10}\cdot5^{20}}{3^{15}\cdot25^{15}}\)\(=\dfrac{\left(3^2\right)^{10}\cdot5^{30}}{3^{15}\cdot\left(5^2\right)^{15}}\)\(=\dfrac{3^{20}\cdot5^{30}}{3^{15}\cdot5^{30}}\)\(=\dfrac{3^{20}}{3^{15}}\)\(=3^4=81\)
\(\text{c)}\) \(\dfrac{390^4}{130^4}=\left(\dfrac{390}{130}\right)^4=3^4=81\)
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