Context: The function $f(x) = ab^{x/n}$ has constants $a$, $b$, $n$ with $b$ and $n$ integers, and $f(5) = 7$ and $f(8) = 189$. We need $f(10)$.

Answer: $1701$

Why: Divide the two known values so $a$ cancels: $\dfrac{f(8)}{f(5)} = b^{8/n - 5/n} = b^{3/n} = \dfrac{189}{7} = 27$. Taking the cube root gives $b^{1/n} = 3$, so each increase of $1$ in $x$ triples the output. From $x = 8$ to $x = 10$ is two steps: $f(10) = 189 \times 3^2 = 189 \times 9 = 1701$.

Verification: With $b^{1/n} = 3$, the condition $f(5) = a \cdot 3^5 = 7$ gives $a = \dfrac{7}{243}$, and then $f(10) = \dfrac{7}{243} \cdot 3^{10} = 7 \cdot 3^5 = 1701$.

Common slips: Multiplying $189$ by $3$ only once (stopping at $x = 9$) gives $567$; treating $27$ as the per-unit growth factor rather than the growth over a $3$-unit gap inflates the answer enormously.