Question

solve these questions
$1) \: {2}^{x} + {4}^{x} = ( {8})^{ \frac{1}{3} } \times (32)^{ \frac{1}{5} } \\ 2) \: ( \frac{2}{3} )^{ \frac{4}{3} } \div ( \frac{2}{3} ) ^{ \frac{5}{3} }$

Answer 1

Answer:-

$(i) \: 2 {}^{x} + 4 {}^{x} = 8 {}^{ \frac{1}{3} } \times 3 {}^{2 \frac{1}{5} } \\ = 2 {}^{x} + (2 {}^{2} ) = 2 {}^{3 \times \frac{1}{3} } \times (2 {}^{5} ) \frac{1}{59} \\ = 2 {}^{x} + 2 {}^{2x} = 2 {}^{3 \times \frac{1}{3} } \times 2 {}^{5 \times \frac{1}{5} } \\ = 2 {}^{ x } + 2 {}^{2x} = 2 \times 2 \\ = 2 {}^{ x} + 2 {}^{2x} = 2 {}^{2} \\ = x + 2 {}^{x} = 2 \\ 3 {}^{x} = 2 \\ x = \frac{2}{3}$

$(ii) \: ( \frac{2}{3} ) {}^{ \frac{4}{3} } \div ( \frac{2}{3} ) {}^{ \frac{5}{3} } \\ = ( \frac{2}{3} ) {}^{ \frac{4}{3 } - \frac{5}{3} } \\ = ( \frac{2}{3} ) {}^{ \frac{ - 1}{3} } \\ =( \frac{3}{2} ) {}^{ \frac{1}{3} } = ( \frac{27}{8} )$

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