Question

2^(x+1)+2^(x-1)-3^(x-1)=3^(x-2)-2^(x-3)+2*3^(x-3)

Answer 1

х=4

Объяснение:

свойства степени:

${a}^{m + n} = {a}^{m} \times {a}^{n} \\ {a}^{m - n} = \frac{ {a}^{m} }{ {a}^{n} } = {a}^{m} \times \frac{1}{ {a}^{n} }$

${2}^{x + 1} + {2}^{x - 1} - {3}^{x - 1} = {3}^{x - 2} - {2}^{x - 3} + 2 \times {3}^{x - 3} \\ {2}^{x + 1} + {2}^{x - 1} + {2}^{x -3 } = {3}^{x - 2} + {3}^{x - 1} + 2 \times {3}^{x - 3}$

${2}^{x} \times {2}^{1} + {2}^{x} \times {2}^{ - 1}+ {2}^{x} \times {2}^{ - 3} = 2 \times {2}^{x} + \frac{1}{2} \times {2}^{x} + \frac{1}{8} \times {2}^{x} = {2}^{x} \times (2 + \frac{1}{2} + \frac{1}{8} ) = {2}^{x} \times \frac{21}{8}$

${3}^{x} \times {3}^{ - 2} + 2 \times {3}^{x} \times {3}^{ - 3} + {3}^{x} \times {3}^{1} = \frac{1}{9} \times {3}^{x} + \frac{2}{27} \times {3}^{x} + \frac{1}{3} \times {3}^{x} = {3}^{x} \times \frac{14}{27}$

${2}^{x} \times \frac{21}{8} = {3}^ {x} \times \frac{14}{27} \: | \: \div ( {3}^{x} \times \frac{14}{27} )$

$\frac{ {2}^{x} \times \frac{21}{8} }{ {3}^{x} \times \frac{14}{27} } = \frac{ {3}^{x} \times \frac{14}{27} }{ {3}^{x} \times \frac{14}{27}} \\ {( \frac{2}{3}) }^{x} \times ( \frac{21}{8} \div \frac{14}{27} ) = 1 \\ {( \frac{2}{3}) }^{x} \times \frac{81}{16} = 1 \\ {( \frac{2}{3}) }^{x} = 1 \div \frac{81}{16}$

${( \frac{2}{3})}^{x} = \frac{ {2}^{4} }{ {3}^{4} } \\ {( \frac{2}{3}) }^{x} = {( \frac{2}{3})}^{4}$

простейшее показательное уравнение

х=4