If $4^{x} - 4^{x - 1} = 24$ , then $(2 x)^{x}$ equals

25

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125

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\sqrt{5}

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25 \sqrt{5}

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Solution

The correct option is D $25 \sqrt{5}$
$4^{x} - 4^{x - 1} = 24$
or $4^{x} (1 - \frac{1}{4}) = 24$
or $4^{x} \frac{3}{4} = 24$
or $4^{x} = \frac{24 \times 4}{3} = 32 = 2^{5}$
or $2^{2 x} = 2^{5}$ or $x = \frac{5}{2}$
or $2 x = 5$ or $(2 x)^{x} = (5)^{5 / 2} = 5^{1} \times 5^{1} \times 5^{1 / 2} = 25 \sqrt{5}$

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