`x \sqrt{y} - y \sqrt{x}`

`= (\sqrt{x})^2 . \sqrt{y} - (\sqrt{y})^2 . \sqrt{x}`

`= \sqrt{x} . \sqrt{y} . (\sqrt{x} - \sqrt{y})`

`= \sqrt{xy} . (\sqrt{x} - \sqrt{y})`

`a) 6x + 4x = 2010`

`=> 10x = 2010`

`=> x = 2010 : 10`

`=> x=201`

Vậy `x=201`

`b) 1+2+...+x = 45`

`=> (x.(x+1))/2 = 45`

`=> x(x+1) = 90`

`=> x^2 + x - 90 = 0`

`=> x^2 + 10x - 9x  -90 = 0`

`=> x(x+10) - 9(x+10) = 0`

`=> (x+10)(x-9) = 0`

`=> [(x+10=0),(x-9=0):}`

`=> [(x=-10),(x=9):}`

Vì `x > 0 => x=9`

Vậy `x=9`

`c) 1+3+5+...+x = 36`

`=> ([(x-1):2+1].(x+1))/2 = 36`

`=> [(x-1):2+1].(x+1) = 72`

`=> (1/2 x - 1/2 + 1) . (x+1) = 72`

`=> (1/2 x + 1/2) (x+1) = 72`

`=> 1/2 . (x+1) (x+1) = 72`

`=> 1/2 (x+1)^2 = 72`

`=> (x+1)^2 = 144`

`=> [(x+1=12),(x+1=-12):}`

`=> [(x=11),(x=-13):}`

Vì `x in NN => x=11`

Vậy `x=11`

`5^(x-2) = 125`

`=> 5^(x-2) = 5^3`

`=> x-2 = 3`

`=> x =5`

Vậy `x=5`

`1+2^1 + 2^2 + 2^3 + ... + 2^2021`

`= (1+2^1) + (2^2 + 2^3) + ... + (2^2020 + 2^2021)`

`= 3 + 2^2 . (1+2) + ... + 2^2020 . (1+2)`

`= 3 + 2^2 . 3 + ... + 2^2020 . 3`

`= 3.(1 + 2^2 + ... + 2^2020) \vdots 3`

`1+2+3+...+n = 1275`

`=> (n(n+1))/2 = 1275`

`=> n(n+1) = 1275 . 2`

`=> n(n+1) = 2550`

Vì `n in NN => n` và `n+1` là hai số tự nhiên liên tiếp

Mà `2550 = 50 . 51`

`=> n=50`

Vậy `n=50`

`@120=2.2.2.3.5 = 2^3 x3x5`             

`@170=2.5.17` 

`@130=2.5.13`             

`@168=2.2.2.3.7 = 2^3 . 3.7`

`@165= 3.5.11`             

`@232=2.2.2.29 = 2^3 . 29`

`x^2 - 25 + y^2 + 2xy

`= x^2 + 2xy + y^2 - 25

`= (x^2 + 2xy + y^2) - 5^2`

`= (x^2 + xy + xy + y) - 5^2`

`= [x(x+y) + y(x+y)] - 5^2`

`= (x+y)(x+y) - 5^2```

`= (x+y)^2 - 5^2`

`= (x+y-5)(x+5+5)`

`2012 = 2.2.503 = 2^2 . 503`