Question

if 4 to the power (2x-1) - 16 to the power of (x-1) = 384 find the value of x​

Answer 1

GIVEN :–

$\\ \implies \bf {4}^{2x - 1} - {16}^{x - 1} = 384$

TO FIND :–

• Value of 'x' = ?

SOLUTION :–

$\\ \implies \bf {4}^{2x - 1} - {16}^{x - 1} = 384$

• We should write this as –

$\\ \implies \bf {4}^{2x - 1} - {( {4}^{2}) }^{x - 1} = 384$

$\\ \implies \bf {4}^{2x - 1} - {4}^{2(x - 1)} = 384$

$\\ \implies \bf {4}^{2x - 1} - {4}^{2x -2} = 384$

• Using identity –

$\\ \implies \bf {a}^{b +c} = {a}^{b} . {a}^{c}$

$\\ \implies \bf {4}^{2x }. {4}^{ - 1} - {4}^{2x}. {4}^{ - 2}= 384$

$\\ \implies \bf {4}^{2x }( {4}^{ - 1} - {4}^{ - 2})= 384$

$\\ \implies \bf {4}^{2x } \left( \dfrac{1}{4} - \dfrac{1}{ {4}^{2} } \right)= 384$

$\\ \implies \bf {4}^{2x } \left( \dfrac{1}{4} - \dfrac{1}{16} \right)= 384$

$\\ \implies \bf {4}^{2x } \left( \dfrac{4 - 1}{16} \right)= 384$

$\\ \implies \bf {4}^{2x } \left( \dfrac{3}{16} \right)= 384$

$\\ \implies \bf {4}^{2x } \left( \dfrac{1}{16} \right)= 128$

$\\ \implies \bf {4}^{2x }= {4}^{2}( 128)$

$\\ \implies \bf {2}^{4x}= {2}^{4} .{2}^{7}$

$\\ \implies \bf {2}^{4x}= {2}^{4 + 7}$

$\\ \implies \bf {2}^{4x}= {2}^{11}$

• Now compare –

$\\ \implies \bf 4x = 11$

$\\ \implies \large { \boxed{\bf x = \dfrac{11}{4}}}$