Đáp án + Giải thích các bước giải : 

Câu 1 : 

`2^3 . 17 - 2^3 . 14`

`= 2^3 . ( 17 - 14 )`

`= 2^3 . 3`

`= 8 . 3`

`= 24`

`_________________________________________`

`64^2 : 16^3 . 199`

`= (4^3)^2 : (4^2)^3 . 199`

`= 4^6 : 4^6 . 199`

`= 1 . 199`

`=199`

`_________________________________________`

`27^3 : 9^4 + 15`

`= (3^3)^3 : (3^2)^4 + 15`

`= 3^9 : 3^8 + 15`

`= 3 + 15`

`= 18`

`_________________________________________`

`17^2 . 5 + 15 . 17`

`= 17 . 17 . 5 + 15 . 17`

`= 17 . ( 17 . 5 + 15 )` 

`= 17 . 100`

`= 1700`

`_________________________________________`

`7 . 2^13 - 4^7 - 10 . 8^4`

`= 7 . 2^13 - (2^2)^7 - 10 . (2^3)^4`

`= 7 . 2^13 - 2^14 - 2 . 5  . 2^12`

`= 7 . 2^13 - 2^13 . 2 - 5 . 2^13`

`= 2^13 . ( 7 - 5 - 2 )`

`= 2^13 . 0 `

`= 0`

`_________________________________________`

`18^3 : 9^3 - 5`

`= (2 . 9)^3 : 9^3 - 5`

`= 2^3 . 9^3 : 9^3 - 5`

`= 2^3 . ( 9^3 : 9^3 ) - 5`

`= 8 . 1 - 5`

`= 3`

Câu 2 : 

`(2^4 . 8^3)/4^6`

`= (2^4 . (2^3)^3/((2^2)^6)`

`= (2^4 . 2^9)/2^12`

`= 2^13/2^12`

`= 2`

`_________________________________________`

`(9^2 . 16^5)/(3^5 . 4^9)`

`= (3^4 . 4^10)/(3^5 . 4^9)`

`= (3^4 . 4)/(3^5)`

`= (3^4 . 4)./(3^4 . 3)`

`= 4/3`

`_________________________________________`

`(9^4 . 2 - 3^6)/(34 . 3^7)`

`= (3^8 . 2 - 3^6 . 1)/(34 . 3^7)`

`= (3^6 . 3 . 3 . 2 - 3^6 . 1)/(34 . 3^7)`

`= (3^6 . (3 . 3 . 2 - 1)/(34 . 3^7)`

`= (3^6 . 17)/(34 . 3^7)`

`_________________________________________`

`(15 . 3^11 + 4 . 27^4)/9^7`

`= (3 . 5 . 3^11 + 4 . (3^3)^4)/((3^2)^7)`

`= (5 . 3^12 + 4 . 3^12)/(3^14)`

`= (3^12 . ( 5 + 4 ))/(3^14)`

`= (3^12 . 3^2)/(3^14)`

`= (3^14)/(3^14)`

`= 1`

Câu 3 : 

Dãy số A có số số hạng là : 

(2010 - 1) : 1 + 1 = 2010 ( số ) 

Dãy A có số cặp số là : 

2010 : 2 = 1005 ( cặp số ) 

`A = 2012 + 2012^2 + 2012^3 + ...... + 2012^2010`

`A = (2012 + 2012^2 ) + (2012^3 + 2012^4 ) + ....... + (2012^2009 + 2012^2010)`

`A = 2012(1 + 2012) + 2012^3(1 + 2012) + .......... + 2012^2009(1 + 2012)`

`A = 2012 xx 2013 + 2012^3 xx 2013 + .......... + 2012^2009 xx 2013`

`A = 2013(2012 + 2012^3 + ......... + 2012^2009)`

`=> A \vdots 2013`

`=> đpcm`

Câu 4 : 

`5^x+1 -5^x=500`

`5^x .5-5^x=500`

`5^x .4=500`

`5^x=500:4`

`5^x=125`

`5^x=5^3`

`=>x=3`

`_________________________`

`2^x +2^x+1 =96`

`2^x +2^x . 2=96`

`2^x .3=96`

`2^x=32`

`2^x=2^5`

`=>x=5`

`_______________________`

`2^x +2^x+3=144`

`2^x +2^x .8=144`

`2^x . 9=144`

`2^x=144:9`

`2^x=16`

`2^x=2^4`

`=>x=4`

`__________________`

`3^2x+2=9^x+3`

`3^2x+2=3^3x+9`

`=>2x+2=3x+9`

`=>3x-2x=9-2`

`=>x=7`

Câu 5 : 

theo đề bài , ta có số cây pải trồng là x ( cây , `x in N , x > 1`)

`x \vdots 10 ; x \vdots 15 => x in BC(10,15)`

ta có : 

`10 = 2 . 5`

`15 = 3 . 5`

`=> BCNN(10,15) = 2 . 3 . 5 = 30`

`=> BC(10,15) = {0 ; 30 ; 60 ; 90 ; 120 ; .....}`

`=> x in {120 ; 150}`

Vậy ..........

Đáp án:

Giải thích các bước giải:

 Câu 3:

A=2012+2012^2+..+2012^2009+2012^2010

A=(2012+2012^2)+..+(2012^2009+2012^2010)

A=2012(1+2012)+...+2012^2009(1+2012)

A=(1+2012)(2012+..+2012^2009)

A=2013(2012+...+2012^2009) chia hết cho 2013

Vậy A chia hết cho 2013

Câu 4:

5^x+1 -5^x=500

5^x .5-5^x=500

5^x .4=500

5^x=500:4

5^x=125

5^x=5^3

=>x=3

2^x +2^x+1 =96

2^x +2^x  .2=96

2^x .3=96

2^x=32

2^x=2^5

=>x=5

2^x +2^x+3=144

2^x +2^x .8=144

2^x .9=144

2^x=144:9

2^x=16

2^x=2^4

=>x=4

3^2x+2=9^x+3

3^2x+2=3^3x+9

=>2x+2=3x+9

=>3x-2x=9-2

=>x=7