`P=(x\sqrt{x}-3)/(x-2\sqrt{x}-3)-(2(\sqrt{x}-3))/(\sqrt{x}+1)+(\sqrt{x}+3)/(3-\sqrt{x})\  (ĐK:x\ge0;x\ne 9)`

`=(x\sqrt{x}-3)/((\sqrt{x}-3)(\sqrt{x}+1))-(2\sqrt{x}-6)/(\sqrt{x}+1)-(\sqrt{x}+3)/(\sqrt{x}-3)`

`=(x\sqrt{x}-3-(2\sqrt{x}-6)(\sqrt{x}-3)-(\sqrt{x}+3)(\sqrt{x}+1))/((\sqrt{x}-3)(\sqrt{x}+1))`

`=(x\sqrt{x}-3-(2x-12\sqrt{x}+18)-(x+4\sqrt{x}+3))/((\sqrt{x}-3)(\sqrt{x}+1))`

`=(x\sqrt{x}-3-2x+12\sqrt{x}-18-x-4\sqrt{x}-3)/((\sqrt{x}-3)(\sqrt{x}+1))`

`=(x\sqrt{x}-3x+8\sqrt{x}-24)/((\sqrt{x}-3)(\sqrt{x}+1))`

`=(x(\sqrt{x}-3)+8(\sqrt{x}-3))/((\sqrt{x}-3)(\sqrt{x}+1))`

`=((\sqrt{x}-3)(\sqrt{x}+8))/((\sqrt{x}-3)(\sqrt{x}+1))`

`=(\sqrt{x}+8)/(\sqrt{x}+1)`

`ĐK:x\ge 0`

`B=x-\sqrt{x}=(x-\sqrt{x}+1/4)-1/4`

`=(\sqrt{x}-(1)/(2))^{2}-1/4\ge -1/4` với mọi `x\ge0`

Dấu ''='' xảy ra `<=>(\sqrt{x}-(1)/(2))^{2}=0<=>\sqrt{x}=1/2<=>x=1/4\  (TMDK)`

Vậy GTNN của `B` là : `-1/4<=>x=1/4`

`(a):` Ta có : `a-b=4=>a=b+4`

Xét tổng các chữ số của `\overline{7a5b1}` :

`7+a+5+b+1=13+a+b=13+b+4+b=17+2b`

Để `\overline{7a5b1}\vdots 3`

Thì : `17+2b\vdots 3`

Vì : `b` là STN có `1` chữ số `=>2b\vdots 2;0\le 2b\le 18`

`=>2b \in {4;10;16}`

`=>b\in {2;5;8}\   (TM)`

`+)\  b=2=>a=2+4=6\  (TM)`

`+)\  b=5=>a=5+4=9\   (TM)`

`+)\   b=8=>a=8+4=12\   (KTM)`

Vậy `(a;b)=(6;2);(9;5)`

`(b):\ 1/2^{2}<(1)/(1.2);(1)/(3^{2})<(1)/(2.3);(1)/(4^{2})<(1)/(3.4);...;(1)/(10^{2})<(1)/(9.10)`

`=>D< (1)/(1.2)+(1)/(2.3)+(1)/(3.4)+(1)/(4.5)+...+(1)/(9.10)`

`=>D< 1-(1)/(2)+(1)/(2)-(1)/(3)+(1)/(3)-(1)/(4)+(1)/(4)-(1)/(5)+...+(1)/(9)-(1)/(10)`

`=>D<1-(1)/(10)<1` (DPCM)

Tổng chiều dài và rộng :

     `86:2=43\  (m)`

Chiều rộng :

    `(43-13):2=15\  (m)`

Chiều dài :

    `15+13=28\   (m)`

Diện tích HCN :

    `28xx15=420\  (m^2)`

`A=1+3+3^{2}+3^{3}+...+3^{99}+3^{100}`

`=>3A=3+3^{2}+3^{3}+3^{4}+...+3^{100}+3^{101}`

`=>3A-A=(3+3^{2}+3^{3}+3^{4}+...+3^{100}+3^{101)-(1+3+3^{2}+3^{3}+...+3^{99}+3^{100})`

`=>2A=3^{101}-1`

`=>A=(3^{101}-1)/(2)`

`x xx6+366=2424`

`x xx6=2424-366`

`x xx6=2058`

`x=2058:6`

`x=343`

Chiều rộng :

    `(25-7):2=9\  (cm)`

Chiều dài :

    `9+7=16\  (cm)`

Chu vi HCN :

    `(16+9)xx2=50\  (cm)`

`A=(42)/(14.20)+(42)/(20.26)+(42)/(26.32)+...+(42)/(80.86)`

`=>(A)/(7)=(6)/(14.20)+(6)/(20.26)+(6)/(26.32)+...+(6)/(80.86)`

`=>(A)/(7)=(20)/(14.20)-(14)/(14.20)+(26)/(20.26)-(20)/(20.26)+(32)/(26.32)-(26)/(26.32)+...+(86)/(80.86)-(80)/(80.86)`

`=>A/7=(1)/(14)-(1)/(20)+(1)/(20)-(1)/(26)+(1)/(26)-(1)/(32)+...+(1)/(80)-(1)/(86)`

`=>A/7=(1)/(14)-(1)/(86)=18/301`

`=>A=(18)/(301).7=18/43`

`(a):\  A=\sqrt{(2-\sqrt{3})^{2}}+2\sqrt{3}`

`=|2-\sqrt{3}|+2\sqrt{3}`

`=2-\sqrt{3}+2\sqrt{3}`

`=2+\sqrt{3}`

`(b):\  B=\sqrt{18}-2\sqrt{50}+3\sqrt{8}+` \(\sqrt[3]{27}\)

`=\sqrt{9}.\sqrt{2}-2.\sqrt{25}.\sqrt{2}+3.\sqrt{4}.\sqrt{2}+3`

`=(3-2.5+3.2).\sqrt{2}+3`

`=3-\sqrt{2}`

`(c):\  C=(4)/(\sqrt{5}-1)-(10)/(\sqrt{5})+(\sqrt{125})/(\sqrt{5})+\sqrt{2}.\sqrt{5/2}`

`=(4(\sqrt{5}+1))/((\sqrt{5}-1)(\sqrt{5}+1))-(10\sqrt{5})/(5)+\sqrt{125/5}+\sqrt{2.(5)/(2)}`

`=(4(\sqrt{5}+1))/(5-1)-2\sqrt{5}+\sqrt{25}+\sqrt{5}`

`=(4(\sqrt{5}+1))/(4)-2\sqrt{5}+5+\sqrt{5}`

`=\sqrt{5}+1-\sqrt{5}+5`

`=6`

`(a):\  P=(1)/(\sqrt{x})-(\sqrt{x})/(\sqrt{x}+3)+(4x-3\sqrt{x}-3)/(\sqrt{x}(\sqrt{x}+3))\  (x>0)`

`=(\sqrt{x}+3-\sqrt{x}^{2}+4x-3\sqrt{x}-3)/(\sqrt{x}(\sqrt{x}+3))`

`=(\sqrt{x}+3-x+4x-3\sqrt{x}-3)/(\sqrt{x}(\sqrt{x}+3))`

`=(3x-2\sqrt{x})/(\sqrt{x}(\sqrt{x}+3))`

`=(3\sqrt{x}-2)/(\sqrt{x}+3)\  (DPCM)`

`(b):\  x=20-6\sqrt{11}=9-2.3.\sqrt{11}+11`

`=(3-\sqrt{11})^{2}`

`=>\sqrt{x}=\sqrt{(3-\sqrt{11})^{2}}=|3-\sqrt{11}|=\sqrt{11}-3`

Thay `\sqrt{x}=\sqrt{11}-3\  (TMDK)` vào `P`, ta được :

`P=(3.(\sqrt{11}-3)-2)/(\sqrt{11}-3+3)=(3\sqrt{11}-11)/(\sqrt{11})`

`=(\sqrt{11}(3-\sqrt{11}))/(\sqrt{11})=3-\sqrt{11}`