1. **State the problem:** Given that $2^a=5$ and $5^b=7$, find the value of $2^{ab+1}$. 2. **Understand the problem:** We want to find $2^{ab+1}$. Notice that $ab+1 = ab + 1$, so we can write: $$2^{ab+1} = 2^{ab} \times 2^1 = 2 \times 2^{ab}$$ 3. **Express $2^{ab}$ in terms of known quantities:** Since $2^a = 5$, raising both sides to the power $b$ gives: $$ (2^a)^b = 5^b $$ which simplifies to: $$ 2^{ab} = 5^b $$ 4. **Use the given value for $5^b$:** From the problem, $5^b = 7$, so: $$ 2^{ab} = 7 $$ 5. **Calculate $2^{ab+1}$:** Recall from step 2: $$ 2^{ab+1} = 2 \times 2^{ab} = 2 \times 7 = 14 $$ **Final answer:** $$\boxed{14}$$