Rút gọn phân số sau :
\(\frac{5^{28}+5^{24}+...+5^4+1}{5^{30}+5^{28}+...+5^2+1}\)
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\(\dfrac{5}{15}=\dfrac{5:5}{15:5}=\dfrac{1}{3}\\ \dfrac{28}{56}=\dfrac{28:28}{56:28}=\dfrac{1}{2}\\ \dfrac{48}{96}=\dfrac{48:48}{96:48}=\dfrac{1}{2}\\ \dfrac{15}{120}=\dfrac{15:15}{120:15}=\dfrac{1}{8}\\ \dfrac{60}{80}=\dfrac{60:20}{80:20}=\dfrac{3}{4}\)
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Đặt \(A=\frac{5^{28}+5^{24}+.........+5^4+1}{5^{30}+5^{28}+5^{26}+.....+5^2+1}=\frac{B}{C}\)
Xét tử \(B=5^{28}+5^{24}+........+5^4+1\)
\(\Rightarrow5^4.B=5^{32}+5^{28}+........+5^8+5^4\)
\(\Rightarrow625.B=5^{32}+5^{28}+........+5^8+5^4\)
\(\Rightarrow625.B-B=624.B=5^{32}-1\)
\(\Rightarrow B=\frac{5^{32}-1}{624}\)
Xét mẫu \(C=5^{30}+5^{28}+5^{26}+........+5^2+1\)
\(\Rightarrow5^2.C=5^{32}+5^{30}+5^{28}+.........+5^4+5^2\)
\(\Rightarrow25.C=5^{32}+5^{30}+5^{28}+........+5^4+5^2\)
\(\Rightarrow25.C-C=24.C=5^{32}-1\)
\(\Rightarrow C=\frac{5^{32}-1}{24}\)
\(\Rightarrow A=\frac{B}{C}=\frac{5^{32}-1}{624}:\frac{5^{32}-1}{24}=\frac{5^{32}-1}{624}.\frac{24}{5^{32}-1}=\frac{24}{624}=\frac{1}{26}\)
Đặt phân số trên là A
\(A=\frac{25^{28}+25^{24}+...+25^4+25^0}{\left(25^{28}+25^{24}+...+25^4+25^0\right)+\left(25^{30}+25^{26}+...+25^6+25^2\right)}\)
\(\frac{1}{A}=\frac{\left(25^{28}+25^{24}+...+25^4+25^0\right)+\left(25^{30}+25^{26}+...+25^6+25^2\right)}{25^{28}+25^{24}+...+25^4+25^0}\)
\(\frac{1}{A}=1+\frac{25^{30}+25^{26}+...+25^6+25^2}{25^{28}+25^{24}+...+25^4+25^0}\)
Đặt \(B=\frac{25^{30}+25^{26}+...+25^6+25^2}{25^{28}+25^{24}+...+25^4+25^0}\)
\(\frac{B}{25^2}=\frac{25^{30}+25^{26}+...+25^6+25^2}{25^{30}+25^{26}+...+25^6+25^2}=1\Rightarrow B=25^2\)
=> \(\frac{1}{A}=1+B=1+25^2\Rightarrow A=\frac{1}{1+25^2}\)
a) \(\frac{2}{5}+\frac{9}{15}=\frac{2}{5}+\frac{3}{5}=\frac{5}{5}=1\)
b) \(\frac{15}{45}+\frac{25}{30}=\frac{1}{3}+\frac{5}{6}=\frac{2}{6}+\frac{5}{6}=\frac{7}{6}\)
c) \(\frac{28}{32}+\frac{45}{72}=\frac{7}{8}+\frac{5}{8}=\frac{12}{8}=\frac{3}{2}\)
d) \(\frac{8}{28}+\frac{5}{30}=\frac{2}{7}+\frac{1}{6}=\frac{12}{42}+\frac{7}{42}=\frac{19}{42}\)
a) \(\frac{2}{5}+\frac{9}{15}\)=\(\frac{2}{5}+\frac{3}{5}=\frac{2+3}{5}=1\)
b)\(\frac{15}{45}+\frac{25}{30}=\frac{1}{3}+\frac{5}{6}=\frac{2}{6}+\frac{5}{6}=\frac{2+5}{6}=\frac{7}{6}\)
c)\(\frac{28}{32}+\frac{45}{72}=\frac{7}{8}+\frac{5}{8}=\frac{7+5}{8}=\frac{12}{8}=\frac{3}{2}\)
d)\(\frac{8}{28}+\frac{5}{30}=\frac{2}{7}+\frac{1}{6}=\frac{12}{42}+\frac{7}{42}=\frac{12+7}{42}=\frac{19}{42}\)