Question

${5}^{0.444x} + {0.3}^{ \sqrt{2} } = {0.2}^{ \sqrt[5]{3} } + 1.2$
precisamos descobrir o valor de x ​

Answer 1

${5}^{0.44 4x} + 0.3 {}^{ \sqrt{2} } = 0.2 {}^{ \sqrt[5]{3} } + 1.2 \\ 5 {}^{0.444x} + \left({  \frac{3}{10} }\right) {}^{ \sqrt{2} } = 0.2 {}^{ \sqrt[5]{3} } + 1.2 \\ {5}^{0.444x} + \left({  \frac{3}{10} }\right) {}^{ \sqrt{2} } = \left({  \frac{2}{10} }\right) {}^{ \sqrt[5]{3} } + \frac{12}{10}$

Simplifique as duas últimas frações

${5}^{0.444x} + \left({  \frac{3}{10} }\right) {}^{ \sqrt{2} } = \left({ \frac{1}{5} }\right) {}^{ \sqrt[5]{3} } + \frac{6}{5} \\ \\ {5}^{0.444x} + \frac{ {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } } = \frac{1}{ {5}^{ \sqrt[5]{3} } } + \frac{6}{5} \\ \\ \frac{ {5}^{0.444x} }{1} + \frac{ {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } } = \frac{1}{ {5}^{ \sqrt[5]{3} } } + \frac{6}{5} \\ \\ \frac{ {10}^{ \sqrt{2} } \times {5}^{0.444x} }{ {10}^{ \sqrt[]{2} } } + \frac{ {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } } = \frac{1}{ {5}^{ \sqrt[5]{3} } } + \frac{6}{5}$

Multiplique a equação por $5 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} }$

$5 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } \times \frac{ {10}^{ \sqrt{2} } \times {5}^{0.444x} }{ {10}^{ \sqrt[]{2} } } + \frac{ {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } } = 5 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } \times \left({  \frac{1}{ {5}^{ \sqrt[5]{3} } } + \frac{6}{5} }\right) \\ \\ 5 \times {\cancel{ {10}^{ \sqrt{2} } } } \times {5}^{ \sqrt[5]{3} } \times \frac{ {10}^{ \sqrt{2} } \times {5}^{0.444x} }{ {\cancel{ {10}^{ \sqrt{2} } } } } + \frac{ {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } } = 5 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } \times \left({  \frac{1}{ {5}^{ \sqrt[5]{3} } } + \frac{6}{5} }\right) \\ \\ 5 \times {5}^{ \sqrt[5]{3} } \times \left({  {10}^{ \sqrt{2} } \times {5}^{0.444x} + {3}^{ \sqrt{2} } }\right) = 5 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } \times \frac{1}{ {5}^{ \sqrt[5]{3} } } + \cancel{5} \times {10}^{ \sqrt{2} } \times 5 {}^{ \sqrt[5]{3} } \times \frac{6}{ \cancel{5}} \\ \\ {5}^{1 + \sqrt[5]{3} } \times \left({  {10}^{ \sqrt{2} } \times {5}^{0.444x} + {3}^{ \sqrt{2} } }\right) = 5 \times {10}^{ \sqrt{2} } + {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } \times 6$

${10}^{ \sqrt{2} } \times {5}^{1 + \sqrt[5]{3} + 0.444x } + {5}^{1 + \sqrt[5]{3} } \times {3}^{ \sqrt{2} } = 5 \times {10}^{ \sqrt{2} } + 6 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } \\ \\ {5}^{1 + \sqrt[5]{3} + 0.444x } = \frac{5 \times {10}^{ \sqrt{2} } + 6 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } - {5}^{1 + \sqrt[5]{3} } \times {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } }$

$log_{5}\left({  {5}^{1 + \sqrt[5]{3} + 0.444x } }\right) = log_{5}\left({  \frac{5 \times {10}^{ \sqrt{2} } + 6 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } - {5}^{1 + \sqrt[5]{3} } \times {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } } }\right) \\ \\ 1 + \sqrt[5]{3} + 0.444x = log_{5}\left({  \frac{5 \times {10}^{ \sqrt{2} } + 6 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } - {5}^{1 + \sqrt[5]{3} } \times {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } } }\right)$

Faça a divisão dos membros da equação por 0,444

$x = log_{5}\left({  \frac{5 \times {10}^{ \sqrt{2} }6 \times {10}^{ \sqrt{2} } \times {5}^{ \sqrt[5]{3} } - {5}^{1 + \sqrt[5]{3} } \times {3}^{ \sqrt{2} } }{ {10}^{ \sqrt{2} } } }\right) \times \frac{250}{111} - \frac{250}{111} - \frac{250 \sqrt[5]{3} }{111} \\ \\\color{green}\begin{gathered} \boxed{\begin{array}{lr}\large\sf\:x≈0,198591 \large \sf \large \sf  \: \end{array}}\end{gathered}$

$\mathcal{Bons \: estudos } \\ \displaystyle\int_ \empty ^ \mathbb{C}     \frac{ - b \: ± \:  \sqrt{ {b}^{2} - 4 \times a \times c } }{2 \times a} d{ t } \boxed{ \boxed{ \mathbb{\displaystyle\Re}\sf{ \gamma  \alpha }\tt{ \pi}\bf{ \nabla}}}$