Bài `3` :
`a) 5^300 = (5^3)^100 = (125)^100`
`3^500 = (3^5)^100 = (243)^100`
Do `125 < 243` nên `(125)^100 < (243)^100`
`=> 5^300 < 3^500`
Vậy `5^300 < 3^500`
`b) 2^24 = (2^3)^8 = 8^8`
`3^16 = (3^2)^8 = 9^8`
Do `8 < 9` nên `8^8 < 9^8`
`=> 2^24 < 3^16`
`c) (-16)^11 = (-2^4)^11 = -2^44`
`(-32)^9 = (-2^5)^9 = -2^45`
Do `45 > 44` nên `-2^45 < -2^44`
`=> - 32^9 < -16^11`
`d) (2^2)^3 = 2^6`
`2^2^3 = 2^8`
`=> (2^2)^3 < 2^2^3`
`e) 2^9^1 = 2^9`
`2^2^3 = 2^6`
`=> 2^9 > 2^6`
`=> 2^9^1 > 2^2^3`
`f) 3 . 24^10 = 3 . (3.8)^10 = 3.3^10.8^10 = 3^11 . (2^3)^10 = 3^11 . 2^30 = 3^11 . (2^2)^15 = 3^11 . 4^15`
Do `3 < 4 ; 11 < 15`
`=> 3^11 < 4^15`
`=> 3^11 . 4^15 < 4^15 . 4^15`
`=> 3 . 24^10 < 4^30`
g) 3/1^2 . 2^2 + 5/2^2 . 3^2 + 7/3^2 . 4^2 + ..... + 19/9^2 . 10^2
`= 3/1.4 + 5/4.9 + 7/9.16 + ...... + 19/81.100`
`= ( 1/1 - 1/4 ) + (1/4 - 1/9) + (1/9 - 1/16) + ...... + (1/81 - 1/100)`
`= 1- 1/4 + 1/4 - 1/9 + 1/9 - 1/16 + ...... + 1/81 - 1/100`
`= 1 - 100 = 99/100`
Có `99/100` và `1`
`=> 99/100` và `100/100`
`=> 99/100 < 100/100`
=> 3/1^2 . 2^2 + 5/2^2 . 3^2 + 7/3^2 . 4^2 + ..... + 19/9^2 . 10^2 < 1
