Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 3.9 AMC 10/12 Find the answer United States

Problem:

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

Your answer

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

Solution

Solution:

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

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