\(a)\left(\dfrac{2}{3}\right)^5=\dfrac{2^5}{3^5}=\dfrac{32}{243}\\ \left(\dfrac{-2}{3}\right)^5=\dfrac{\left(-2\right)^5}{3^5}=\dfrac{-32}{243}\\ \left(-1\dfrac{3}{4}\right)^2=\left(-\dfrac{7}{4}\right)^2=\dfrac{\left(-7\right)^2}{4^2}=\dfrac{49}{16}\\ \left(-0,1\right)^4=\left(-\dfrac{1}{10}\right)^4=\dfrac{\left(-1\right)^4}{10^4}=\dfrac{1}{10000}\)

\(b)\dfrac{90^3}{15^3}=\left(\dfrac{90}{15}\right)^3=6^3=216\\ \dfrac{790^4}{79^4}=\left(\dfrac{790}{79}\right)^4=10^4=10000\\ \dfrac{3^2}{15^2}=\left(\dfrac{3}{15}\right)^2=\left(\dfrac{1}{5}\right)^2=\dfrac{1^2}{5^2}=\dfrac{1}{25}\\ \dfrac{\left(-\dfrac{1}{2}\right)^n}{\left(-\dfrac{1}{2}\right)^{n-1}}=\left(-\dfrac{1}{2}\right)^{n-\left(n-1\right)}=\left(-\dfrac{1}{2}\right)^{n-n+1}=\left(-\dfrac{1}{2}\right)\)

a: \(\dfrac{9}{70}=\dfrac{9\cdot3}{70\cdot3}=\dfrac{27}{210};\dfrac{5}{42}=\dfrac{5\cdot5}{42\cdot5}=\dfrac{25}{210}\)

mà 27>25

nên \(\dfrac{9}{70}>\dfrac{5}{42}\)

b: \(\dfrac{-4}{27}=\dfrac{-4\cdot7}{27\cdot7}=\dfrac{-28}{189};\dfrac{15}{-63}=\dfrac{-15}{63}=\dfrac{-15\cdot3}{63\cdot3}=\dfrac{-45}{189}\)

mà -28>-45

nên \(-\dfrac{4}{27}>-\dfrac{15}{63}\)

a: \(\dfrac{24}{35}=\dfrac{24\cdot6}{35\cdot6}=\dfrac{144}{210};\dfrac{19}{30}=\dfrac{19\cdot7}{30\cdot7}=\dfrac{133}{210}\)

b: \(\dfrac{-9}{21}=\dfrac{-9:3}{21:3}=\dfrac{-3}{7};\dfrac{27}{-63}=\dfrac{27:\left(-9\right)}{-63:\left(-9\right)}=\dfrac{-3}{7}\)

Do đó: \(\dfrac{-9}{21}=\dfrac{27}{-63}\)

a: \(\dfrac{7}{8}=\dfrac{7\cdot3}{8\cdot3}=\dfrac{21}{24};\dfrac{11}{12}=\dfrac{11\cdot2}{12\cdot2}=\dfrac{22}{24}\)

mà 21<22

nên \(\dfrac{7}{8}< \dfrac{11}{12}\)

b: \(\dfrac{-5}{8}=\dfrac{-5\cdot5}{8\cdot5}=\dfrac{-25}{40};\dfrac{7}{-10}=\dfrac{-7}{10}=\dfrac{-7\cdot4}{10\cdot4}=\dfrac{-28}{40}\)

mà -25>-40

nên \(-\dfrac{5}{8}>\dfrac{7}{-10}\)

a: \(\dfrac{998}{555}=\dfrac{555+443}{555}=1+\dfrac{443}{555}\)

\(\dfrac{999}{556}=\dfrac{556+443}{556}=1+\dfrac{443}{556}\)

mà \(\dfrac{443}{555}>\dfrac{443}{556}\)

nên \(\dfrac{998}{555}>\dfrac{999}{556}\)

b: \(\dfrac{315}{380}=1-\dfrac{65}{380};\dfrac{316}{381}=1-\dfrac{65}{381}\)

Ta có: 380<381

=>\(\dfrac{65}{380}>\dfrac{65}{381}\)

=>\(-\dfrac{65}{380}< -\dfrac{65}{381}\)

=>\(-\dfrac{65}{380}+1< -\dfrac{65}{381}+1\)

=>\(\dfrac{315}{380}< \dfrac{316}{381}\)

=>\(-\dfrac{315}{380}>-\dfrac{316}{381}\)

a: \(\dfrac{15}{21}=\dfrac{15:3}{21:3}=\dfrac{5}{7};\dfrac{21}{49}=\dfrac{21:7}{49:7}=\dfrac{3}{7}\)

mà 5>3

nên \(\dfrac{15}{21}>\dfrac{21}{49}\)

b: \(\dfrac{-19}{49}=\dfrac{-19\cdot47}{49\cdot47}=\dfrac{-893}{2303}\)

\(\dfrac{-23}{47}=\dfrac{-23\cdot49}{49\cdot47}=\dfrac{-1127}{2303}\)

mà -893>-1127

nên \(-\dfrac{19}{49}>-\dfrac{23}{47}\)

a: \(\dfrac{-11}{6}< -1\)

\(-1=\dfrac{-9}{9}< \dfrac{8}{-9}\)

Do đó: \(\dfrac{-11}{6}< \dfrac{8}{-9}\)

b: \(-\dfrac{25}{20}< 0\)

\(0< \dfrac{20}{25}\)

Do đó: \(-\dfrac{25}{20}< \dfrac{20}{25}\)

Bài 3:

\(A=75\left(4^{2004}+4^{2003}+...+4^2+4+1\right)+25\)

Đặt: \(B=4^{2004}+4^{2003}+...+4^2+4+1\)

\(4B=4^{2005}+4^{2004}+...+4^3+4^2+4\\ 4B-B=\left(4^{2005}+4^{2004}+...+4^3+4^2+4\right)-\left(4^{2004}+4^{2003}+...+4^2+4+1\right)\\ 3B=4^{2005}-1\\ B=\dfrac{4^{2005}-1}{3}\)

\(=>A=75\cdot\dfrac{4^{2005}-1}{3}+25\\ =25\left(4^{2005}-1\right)+25\\ =25\cdot\left(4^{2005}-1+1\right)\\ =25\cdot4^{2005}\\ =25\cdot4\cdot4^{2004}\\ =100\cdot4^{2004}\) 

=> A chia hết cho 100 

Bài 1:

a: \(\dfrac{5\cdot3^{11}+4\cdot3^{17}}{3^9\cdot5^2-3^9\cdot2^3}=\dfrac{3^{11}\cdot\left(5+4\cdot3^6\right)}{3^9\left(5^2-2^3\right)}\)

\(=3^2\cdot\dfrac{5+4\cdot729}{25-8}=3^2\cdot\dfrac{2921}{17}=\dfrac{26289}{17}\)

b: \(\dfrac{1}{2\cdot5}+\dfrac{1}{5\cdot8}+...+\dfrac{1}{47\cdot50}\)

\(=\dfrac{1}{3}\left(\dfrac{3}{2\cdot5}+\dfrac{3}{5\cdot8}+...+\dfrac{3}{47\cdot50}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{2}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{8}+...+\dfrac{1}{47}-\dfrac{1}{50}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{2}-\dfrac{1}{50}\right)=\dfrac{1}{3}\cdot\dfrac{24}{50}=\dfrac{24}{150}=\dfrac{8}{50}=\dfrac{4}{25}\)

c: \(1+\dfrac{2}{6}+\dfrac{2}{12}+...+\dfrac{2}{9900}\)

\(=\dfrac{2}{2}+\dfrac{2}{6}+...+\dfrac{2}{9900}\)

\(=2\left(\dfrac{1}{2}+\dfrac{1}{6}+...+\dfrac{1}{9900}\right)\)

\(=2\left(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{99\cdot100}\right)\)

\(=2\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}\right)\)

\(=2\left(1-\dfrac{1}{100}\right)=2\cdot\dfrac{99}{100}=\dfrac{99}{50}\)

d: \(\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)\cdot...\cdot\left(\dfrac{1}{99^2}-1\right)\)

\(=\left(\dfrac{1}{2}-1\right)\left(\dfrac{1}{3}-1\right)\cdot...\cdot\left(\dfrac{1}{99}-1\right)\left(\dfrac{1}{2}+1\right)\cdot\left(\dfrac{1}{3}+1\right)\cdot...\cdot\left(\dfrac{1}{99}+1\right)\)

\(=\dfrac{-1}{2}\cdot\dfrac{-2}{3}\cdot...\cdot\dfrac{-98}{99}\cdot\dfrac{3}{2}\cdot\dfrac{4}{3}\cdot...\cdot\dfrac{100}{99}\)

\(=\dfrac{1}{99}\cdot\dfrac{100}{2}=\dfrac{50}{99}\)

\(a)3^{2x-1}+2\cdot9^{x-1}=405\\ =>3^{2x-1}+2\cdot\left(3^2\right)^{x-1}=405\\ =>3^{2x-1}+2\cdot3^{2x-2}=405\\ =>3^{2x-2}\cdot\left(3+2\right)=405\\ =>3^{2x-2}\cdot5=405\\ =>3^{2x-2}=\dfrac{405}{5}=81\\ =>3^{2x-2}=3^4\\ =>2x-2=4\\ =>2x=4+2=6\\ =>x=\dfrac{6}{2}\\ =>x=3\)

\(b)\left(\dfrac{1}{3}\right)^{x-1}+5\left(\dfrac{1}{3}\right)^{x+1}=\dfrac{14}{9^3}\\ =>\left(\dfrac{1}{3}\right)^{x-1}\left(1+5\cdot\dfrac{1}{3^2}\right)=\dfrac{14}{729}\\ =>\left(\dfrac{1}{3}\right)^{x-1}\cdot\dfrac{14}{9}=\dfrac{14}{729}\\ =>\left(\dfrac{1}{3}\right)^{x-1}=\dfrac{14}{729}:\dfrac{14}{9}\\ =>\left(\dfrac{1}{3}\right)^{x-1}=\dfrac{9}{729}=\dfrac{1}{81}\\ =>\left(\dfrac{1}{3}\right)^{x-1}=\left(\dfrac{1}{3}\right)^4\\ =>x-1=4\\ =>x=1+4\\ =>x=5\)

\(c)\dfrac{3}{5}\left(3x^3-\dfrac{8}{9}\right)-\dfrac{1}{2}\left(\dfrac{3}{2}-1\right)=-\dfrac{1}{4}\\ =>\dfrac{3}{5}\left(3x^3-\dfrac{8}{9}\right)-\dfrac{1}{2}\cdot\dfrac{1}{2}=-\dfrac{1}{4}\\ =>\dfrac{3}{5}\left(3x^3-\dfrac{8}{9}\right)-\dfrac{1}{4}=-\dfrac{1}{4}\\ =>\dfrac{3}{5}\left(3x^3-\dfrac{8}{9}\right)=0\\ =>3x^3-\dfrac{8}{9}=0\\ =>3x^3=\dfrac{8}{9}\\ =>x^3=\dfrac{8}{9}:3=\dfrac{8}{27}\\ =>x^3=\left(\dfrac{2}{3}\right)^3\\ =>x=\dfrac{2}{3}\)

a: \(3^{2x-1}+2\cdot9^{x-1}=405\)

=>\(\dfrac{3^{2x}}{3}+2\cdot3^{2x-2}=405\)

=>\(\dfrac{1}{3}\cdot3^{2x}+2\cdot3^{2x}\cdot\dfrac{1}{9}=405\)

=>\(3^{2x}\cdot\left(\dfrac{1}{3}+\dfrac{2}{9}\right)=405\)

=>\(3^{2x}\cdot\dfrac{5}{9}=405\)

=>\(3^{2x}=405:\dfrac{5}{9}=405\cdot\dfrac{9}{5}=81\cdot9=3^6\)

=>2x=6

=>x=3

b: \(\left(\dfrac{1}{3}\right)^{x-1}+5\cdot\left(\dfrac{1}{3}\right)^{x+1}=\dfrac{14}{9^3}\)

=>\(\left(\dfrac{1}{3}\right)^x\cdot3+5\cdot\left(\dfrac{1}{3}\right)^x\cdot\dfrac{1}{3}=\dfrac{14}{9^3}\)

=>\(\left(\dfrac{1}{3}\right)^x\cdot\left(3+\dfrac{5}{3}\right)=\dfrac{14}{9^3}\)

=>\(\left(\dfrac{1}{3}\right)^x=\dfrac{14}{3^6}:\dfrac{14}{3}=\dfrac{3}{3^6}=\dfrac{1}{3^5}\)

=>x=5

c: \(\dfrac{3}{5}\left(3x^3-\dfrac{8}{9}\right)-\dfrac{1}{2}\left(\dfrac{3}{2}-1\right)=-\dfrac{1}{4}\)

=>\(\dfrac{9}{5}x^3-\dfrac{24}{45}-\dfrac{1}{2}\cdot\dfrac{1}{2}+\dfrac{1}{4}=0\)

=>\(\dfrac{9}{5}x^3=\dfrac{24}{45}=\dfrac{8}{15}\)

=>\(x^3=\dfrac{8}{15}:\dfrac{9}{5}=\dfrac{8}{15}\cdot\dfrac{5}{9}=\dfrac{40}{135}=\dfrac{8}{27}=\left(\dfrac{2}{3}\right)^3\)

=>\(x=\dfrac{2}{3}\)

d: \(\dfrac{7}{x}+\dfrac{4}{5\cdot9}+\dfrac{4}{9\cdot13}+...+\dfrac{4}{41\cdot45}=\dfrac{29}{45}\)

=>\(\dfrac{7}{x}+\dfrac{1}{5}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{13}+...+\dfrac{1}{41}-\dfrac{1}{45}=\dfrac{29}{45}\)

=>\(\dfrac{7}{x}+\dfrac{9}{45}-\dfrac{1}{45}=\dfrac{29}{45}\)

=>\(\dfrac{7}{x}=\dfrac{29}{45}-\dfrac{8}{45}=\dfrac{21}{45}=\dfrac{7}{15}\)

=>x=15

e: \(\dfrac{1}{3\cdot5}+\dfrac{1}{5\cdot7}+...+\dfrac{1}{\left(2x+1\right)\left(2x+3\right)}=\dfrac{5}{31}\)

=>\(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{\left(2x+1\right)\left(2x+3\right)}=\dfrac{10}{31}\)

=>\(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{2x+1}-\dfrac{1}{2x+3}=\dfrac{10}{31}\)

=>\(\dfrac{1}{3}-\dfrac{1}{2x+3}=\dfrac{10}{31}\)

=>\(\dfrac{1}{2x+3}=\dfrac{1}{3}-\dfrac{10}{31}=\dfrac{1}{93}\)

=>2x+3=93

=>2x=90

=>x=45

a; (2\(x\) - 1)² = 49

     (2\(x\) - 1)² = 7²

     \(\left[{}\begin{matrix}2x-1=-7\\2x-1=7\end{matrix}\right.\)

       \(\left[{}\begin{matrix}2x=-7+1\\2x=7+1\end{matrix}\right.\)

       \(\left[{}\begin{matrix}2x=-6\\2x=8\end{matrix}\right.\)

         \(\left[{}\begin{matrix}x=-\dfrac{6}{2}\\x=\dfrac{8}{2}\end{matrix}\right.\)

        \(\left[{}\begin{matrix}x=-3\\x=4\end{matrix}\right.\)

Vậy \(x\) \(\in\) {- 3; 4} 

b;  (5\(x-3\))² - (4\(x\) - 7)² = 0

    (5\(x\) - 3 - 4\(x\) + 7)(5\(x\) - 3 + 4\(x\) - 7) = 0

   (\(x\) + 4)(9\(x\) - 10) = 0

    \(\left[{}\begin{matrix}x+4=0\\9x-10=0\end{matrix}\right.\)

     \(\left[{}\begin{matrix}x=-4\\x=\dfrac{10}{9}\end{matrix}\right.\)

Vậy  \(x\) \(\in\) {- 4; \(\dfrac{10}{9}\)}