Question

if {(4)^2x-1)} -{(16)^x-1}=384 then find the value of x

Answer 1

${4}^{2x - 1} - {16}^{x - 1} = 384\\{4}^{2x - 1} - {4}^{2x - 2} = 384 \\ \frac{ {4}^{2x} }{ {4}^{1} } - \frac{ {4}^{2x} }{ {4}^{2} } = 384 \\ {4}^{2x} ( \frac{1}{4} - \frac{1}{16} ) = 384 \\ 4 ^{2x} ( \frac{4 - 1}{16} ) = 384 \\ {4}^{2x} ( \frac{3}{16} ) = 384 \\ {4}^{2x} = 384 \times \frac{16}{3} \\ {4}^{2x} = 128 \times 16 \\ {4}^{2x} = 64 \times 2 \times 16 \\ {4}^{2x} = {4}^{3} \times {4}^{ \frac{1}{2} } \times {4}^{2} \\ {4}^{2x} = 4 ^{3 + \frac{1}{2} + 2} \\ 4 ^{2x} = 4 ^{ \frac{6 + 1 + 4}{2} } \\ {4}^{2x} = 4 ^{ \frac{11}{2} } \\ 2x = \frac{11}{2} \\ x = \frac{11}{2} \times \frac{1}{2} \\ x = \frac{11}{4}$