7.209. $49^{1+\sqrt{x-2}}-344 \cdot 7^{\sqrt{x-2}}=-7$.

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7.209. $49^{1+\sqrt{x-2}}-344 \cdot 7^{\sqrt{x-2}}=-7$.
Solution.

Domain of definition: $x \geq 2$.

Rewrite the equation as $49 \cdot 7^{2 \sqrt{x-2}}-344 \cdot 7^{\sqrt{x-2}}+7=0$. Solving it as a quadratic equation in terms of $7^{\sqrt{x-2}}$, we get $\left(7^{\sqrt{x-2}}\right)=7^{-2}$ or $\left(7^{\sqrt{x-2}}\right)_{2}=7$, from which $(\sqrt{x-2})_{1}=-2$ (no solutions), or $(\sqrt{x-2})_{2}=1, x_{2}=3$.

Answer: 3.