\(4x^2+1=4x\\ < =>4x^2-4x+1=0\\ < =>\left(2x\right)^2-2.2x.1+1^2=0\\ < =>\left(2x-1\right)^20\\ < =>2x-1=0\\ < =>x=\dfrac{1}{2}\)

\(Từ\ trang\ 1->\ trang\ 9,\ có : (9-1):1+1=9\ số, \ hay\ có : 9x1=9\ (chữ số)\)

\(Từ\ trang\ 10->\ trang\ 99,\ có : (99-10):1+1=90\ số,\ hay\ có : 90x2=180\ (chữ số)\)\(Từ\ trang\ 100->\ trang\ 145, có : (145-100):1+1=46\ số,\ hay\ có : 46x3=138\ (chữ số)\)

\(=>\ Minh\ đã\ dùng\ : 9+180+138=327\ (chữ số)\)

\(D=4+4^2+4^3+...+4^{20}\\ =>4D=4^2+4^3+4^4+...+4^{21}\\ =>4D-D=4^2+4^3+4^4+...+4^{21}-\left(4+4^2+4^3+...+4^{20}\right)\\ =>3D=4^{21}-4\\ =>D=\dfrac{4^{21}-4}{3}\\\)

\(M=1+2+2^2+...+2^{40}\\ =>2M=2+2^2+2^3+...+2^{41}\\ =>2M-M=2+2^2+2^3+...+2^{41}-\left(1+2+2^2+...+2^{40}\right)\\ =>M=2^{41}-1\)

\(\dfrac{x-2}{76}+\dfrac{x-4}{74}=\dfrac{x-6}{72}+\dfrac{x-8}{70}\\ =>\left(\dfrac{x-2}{76}-1\right)+\left(\dfrac{x-4}{74}-1\right)=\left(\dfrac{x-6}{72}-1\right)+\left(\dfrac{x-8}{70}-1\right)\\ =>\dfrac{x-78}{76}+\dfrac{x-78}{74}-\dfrac{x-78}{72}-\dfrac{x-78}{70}=0\\ =>\left(x-78\right)\left(\dfrac{1}{76}+\dfrac{1}{74}-\dfrac{1}{72}-\dfrac{1}{70}\right)=0\\ =>x-78=0\\ =>x=78\)

\((Vì\ : \ 1/76+1/74-1/72-1/70<0)\)

\(\dfrac{x-10}{22}+\dfrac{x-8}{24}+\dfrac{x-6}{26}=3\\ =>\left(\dfrac{x-10}{22}-1\right)+\left(\dfrac{x-8}{24}-1\right)+\left(\dfrac{x-6}{26}-1\right)=0\\ =>\dfrac{x-32}{22}+\dfrac{x-32}{24}+\dfrac{x-32}{26}=0\\ =>\left(x-32\right)\left(\dfrac{1}{22}+\dfrac{1}{24}+\dfrac{1}{26}\right)=0\\ =>x-32=0\\ =>x=32\)

\(A=\dfrac{2-\sqrt{3}}{1+\sqrt{4+2\sqrt{3}}}+\dfrac{2+\sqrt{3}}{1-\sqrt{4-2\sqrt{3}}}\\ =\dfrac{2-\sqrt{3}}{1+\sqrt{3+2\sqrt{3}+1}}+\dfrac{2+\sqrt{3}}{1-\sqrt{3-2\sqrt{3}+1}}\\ =\dfrac{2-\sqrt{3}}{1+\sqrt{\sqrt{3}^2+2.\sqrt{3}.1+1^2}}+\dfrac{2+\sqrt{3}}{1-\sqrt{\sqrt{3}^2-2.\sqrt{3}.1+1^2}}\)

\(=\dfrac{2-\sqrt{3}}{1+\sqrt{\left(\sqrt{3}+1\right)^2}}+\dfrac{2+\sqrt{3}}{1-\sqrt{\left(\sqrt{3}-1\right)^2}}\\ =\dfrac{2-\sqrt{3}}{1+\left|\sqrt{3}+1\right|}+\dfrac{2+\sqrt{3}}{1-\left|\sqrt{3}-1\right|}\\ =\dfrac{2-\sqrt{3}}{1+\sqrt{3}+1}+\dfrac{2+\sqrt{3}}{1-\left(\sqrt{3}-1\right)}\\ =\dfrac{2-\sqrt{3}}{2+\sqrt{3}}+\dfrac{2+\sqrt{3}}{2-\sqrt{3}}\\ =\dfrac{\left(2-\sqrt{3}\right)^2+\left(2+\sqrt{3}\right)^2}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}\\ =\dfrac{4-4\sqrt{3}+3+4+4\sqrt{3}+3}{2^2-\sqrt{3}^2}\\ =\dfrac{14}{4-3}=14\)

\(4x^2+20x+25=\left(2x\right)^2+2.2x.5+5^2=\left(2x+5\right)^2\)

\(x^2-4x+5=\left(x^2-4x+4\right)+1=\left(x-2\right)^2+1>=1>0\forall x\)

\(e)\)\(\dfrac{\sqrt{a}-a}{\sqrt{a}-1}=\dfrac{-\sqrt{a}\left(-1+\sqrt{a}\right)}{-1+\sqrt{a}}=-\sqrt{a}\left(ĐK:a>=0;a\ne1\right)\)

\(g)\)\(\dfrac{a-b}{\sqrt{a}-\sqrt{b}}=\dfrac{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}-\sqrt{b}}=\sqrt{a}+\sqrt{b}\left(ĐK:a,b>=0;a\ne b\right)\)

\(i)\)\(\dfrac{x\sqrt{x}-y\sqrt{y}+x\sqrt{y}-y\sqrt{x}}{x+y+2\sqrt{xy}}\\ =\dfrac{x\left(\sqrt{x}+\sqrt{y}\right)-y\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)^2}\\ =\dfrac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-y\right)}{\left(\sqrt{x}+\sqrt{y}\right)^2}\\ =\dfrac{x-y}{\sqrt{x}+\sqrt{y}}\\ =\dfrac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+\sqrt{y}}=\sqrt{x}-\sqrt{y}\left(ĐK:x,y>=0;\sqrt{x}+\sqrt{y}\ne0\right)\)\(\dfrac{a+b+2\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=\dfrac{\left(\sqrt{a}+\sqrt{b}\right)^2}{\sqrt{a}+\sqrt{b}}=\sqrt{a}+\sqrt{b}\left(ĐK:a,b>=0;\sqrt{a}+\sqrt{b}\ne0\right)\)

\(a)\)\(\dfrac{2-\sqrt{2}}{\sqrt{2}}=\dfrac{\sqrt{2}\left(\sqrt{2}-1\right)}{\sqrt{2}}=\sqrt{2}-1\)

\(b)\)\(\dfrac{\sqrt{2}+\sqrt{3}}{2+\sqrt{6}}=\dfrac{\sqrt{2}+\sqrt{3}}{\sqrt{2}\left(\sqrt{2}+\sqrt{3}\right)}=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}\)

\(c)\)\(\dfrac{a+\sqrt{a}}{\sqrt{a}}=\dfrac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}}=\sqrt{a}+1\left(ĐK:a>0\right)\)

\(d)\)\(\dfrac{3+\sqrt{3}}{1+\sqrt{3}}=\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}+1}=\sqrt{3}\)