Ace Algebra 2 Honors: Conquer the 2026 Challenge with Confidence!

This question hinges on representing powers with a common base and equating the exponents. Rewrite both sides with base 2: 4^x = (2^2)^x = 2^{2x} and 64^{-x} = (2^6)^{-x} = 2^{-6x}. Since 2^{2x} = 2^{-6x} and the base 2 is positive and not 1, the exponents must be equal: 2x = -6x, which gives 8x = 0 and x = 0.

Check: 4^0 = 1 and 64^0 = 1, so both sides match. Another quick route is to multiply both sides by 64^x to get (4·64)^x = 1, i.e., 256^x = 1, which also leads to x = 0 since 256 ≠ 1. Therefore, the solution is x = 0.