HMMT November

Algebra Difficulty 4.2 AIME Find the answer

Let $x$ be a real number such that $2^{x}=3$ . Determine the value of $4^{3 x+2}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We have $4^{3 x+2}=4^{3 x} \cdot 4^{2}=\left(2^{2}\right)^{3 x} \cdot 16=2^{6 x} \cdot 16=\left(2^{x}\right)^{6} \cdot 16=3^{6} \cdot 16=11664$

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