Ta có: \(\frac{1}{2.2}.\frac{2}{2.3}.\frac{3}{2.4}.\frac{4}{2.5}.\frac{5}{2.6}.......\frac{30}{2.31}.\frac{31}{64}=4^x\)
\(\frac{1}{2^{30}.64}=4^x\Leftrightarrow4^x.2^{30}.2^6=1\)
\(\Leftrightarrow2^{2x+36}2^0\)
\(\Leftrightarrow2x+36=0\)
\(\Leftrightarrow2x=-36\)
\(\Leftrightarrow x=-18\)
Vậy ........
$4^x.64=1$\(\frac{1}{4}.\frac{2}{6}.\frac{3}{8}.\frac{4}{10}.\frac{5}{12}.....\frac{30}{62}.\frac{31}{64}=4^x\)
\(\Leftrightarrow\frac{1}{2.2}.\frac{2}{2.3}.\frac{3}{2.4}.\frac{4}{2.5}.\frac{5}{2.6}.....\frac{30}{2.31}.\frac{31}{2.32}=4^x\)
\(\Leftrightarrow\frac{1}{2}\left(\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}.\frac{5}{6}.....\frac{30}{31}.\frac{31}{32}\right)=4^x\)
\(\Leftrightarrow\frac{1}{2}.\frac{1.2.3.4.5.....30.31}{2.3.4.5.6.....31.32}=4^x\)
\(\Leftrightarrow\frac{1}{2}.\frac{1}{32}=4^x\)
\(\Leftrightarrow4^x=\frac{1}{64}\)
\(\Leftrightarrow4^x.64=1\)
\(\Leftrightarrow4^x.4^3=1\Leftrightarrow4^{x+3}=4^0\Leftrightarrow x+3=0\Leftrightarrow x=-3\)
Vậy x = -3
\(\frac{1}{4}.\frac{2}{6}.\frac{3}{8}.\frac{4}{10}.\frac{5}{12}...\frac{30}{62}.\frac{31}{64}=4^x\)
\(\Leftrightarrow\frac{1}{2.2}.\frac{2}{2.3}.\frac{3}{2.4}.\frac{4}{2.5}.\frac{5}{2.6}...\frac{30}{2.31}.\frac{31}{2.32}=4^x\)
\(\Leftrightarrow\frac{1.2.3.4.5...30.31}{2.2.2.3.2.4.2.5.2.6...2.31.2.32}=4^x\)
\(\Leftrightarrow\frac{2.3.4.5...30.31}{2^{31}.32.\left(2.3.4.5...31\right)}=4^x\)
\(\Leftrightarrow\frac{1}{2^{31}.2^5}=4^x\)
\(\Leftrightarrow\frac{1}{2^{36}}=4^x\)
\(\Leftrightarrow\frac{1}{4^{18}}=4^x\)
\(\Leftrightarrow x=-18\)
\(\frac{1}{4}\cdot\frac{2}{6}\cdot\frac{3}{8}\cdot\frac{4}{10}\cdot\cdot\cdot\frac{30}{62}\cdot\frac{31}{64}=4^x\)
\(\Leftrightarrow\frac{1}{2.2}\cdot\frac{2}{2.3}\cdot\frac{3}{2.4}\cdot\cdot\cdot\cdot\frac{30}{2.31}\cdot\frac{31}{2.32}=4^x\)
\(\Leftrightarrow\frac{1.2.3.4....30.31}{2^{30}.\left(2.3.4.5....32\right)}=4^x\)
\(\Leftrightarrow\frac{1}{2^{30}.32}=4^x\Leftrightarrow\frac{1}{2^{36}}=4^x\)
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