Question

If 4^2x-1 - 16^x-1 = 384, find the value of x.

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Answer 1

$Answer \: \\ \\ Given \: \: Question \: \: Is \: \: \\ \\ 4 {}^{(2x - 1)} - 16 {}^{(x - 1)} = 384 \\ \\ 4 {}^{2x} \times 4 {}^{ - 1} - 4 {}^{2x} \times 4 {}^{ - 2} = 384 \\ \\ 4 {}^{2x} ( 4 {}^{ - 1} - 4 {}^{ - 2} ) = 384 \\ \\ 4 {}^{2x} ( \frac{1}{4} - \frac{1}{16} ) = 384 \\ \\ 4 {}^{2x} ( \frac{16 - 4}{16 \times 4} ) = 384 \\ \\ 4 {}^{2x} ( \frac{12}{64} ) = 384 \\ \\ 4 {}^{2x} = \frac{384 \times 64}{12} \\ \\ 4 {}^{2x} = 32 \times 64 \\ \\ 4 {}^{2x} = 2 {}^{5} \times 2 {}^{6} \\ \\ 4 {}^{2x} = 2 {}^{11} \\ \\ 2 {}^{4x} = 2 {}^{11} \\ compare \: powers \: of \: 2 \: we \: have \\ \\ 4x = 11 \\ \\ x = \frac{11}{4}$

Answer 2

Answer:

→ $\Huge x = \frac{11}{4}$

Step-by-step explanation:

$\begin{lgathered}\\ \\ 4 {}^{(2x - 1)} - 16 {}^{(x - 1)} = 384 \\ \\ 4 {}^{2x} \times 4 {}^{ - 1} - 4 {}^{2x} \times 4 {}^{ - 2} = 384 \\ \\ 4 {}^{2x} ( 4 {}^{ - 1} - 4 {}^{ - 2} ) = 384 \\ \\ 4 {}^{2x} ( \frac{1}{4} - \frac{1}{16} ) = 384 \\ \\ 4 {}^{2x} ( \frac{16 - 4}{16 \times 4} ) = 384 \\ \\ 4 {}^{2x} ( \frac{12}{64} ) = 384 \\ \\ 4 {}^{2x} = \frac{384 \times 64}{12} \\ \\ 4 {}^{2x} = 32 \times 64 \\ \\ 4 {}^{2x} = 2 {}^{5} \times 2 {}^{6} \\ \\ 4 {}^{2x} = 2 {}^{11} \\ \\ 2 {}^{4x} = 2 {}^{11} \\ \\ Compare \: Powers \: Of \: 2 \: We \: Have \\ \\ 4x = 11 \\ \\ x = \frac{11}{4}\end{lgathered}$