AP DSC School Assistant Math Model Test 2

Concept:This problem uses the method of equating exponents after expressing all variables in terms of a common base. Given three quantities equal, we set them equal to a common constant, then substitute into the given relation to find the required exponent.Explanation:Let $a^{x}=b^{y}=c^{z}=k$. Then we have:$a = k^{1/x},\quad b = k^{1/y},\quad c = k^{1/z}$.The second condition is $b^{2}=ac$. Substituting the expressions:$k^{2/y} = k^{1/x} \cdot k^{1/z} = k^{1/x + 1/z}$.Since the bases are the same, the exponents must be equal:$\frac{2}{y} = \frac{1}{x} + \frac{1}{z}$.Combine the right-hand side:$\frac{2}{y} = \frac{z+x}{xz}$.Take the reciprocal of both sides (or cross‑multiply):$\frac{y}{2} = \frac{xz}{x+z}$.Multiply both sides by 2 to isolate $y$:$y = \frac{2xz}{x+z}$.This matches option D.Answer:Option D: $\frac{2xz}{z+x}$.