1. **State the problem:** Simplify the expression $2^{k+1} + 2^{k+1}$. 2. **Recall the rule for adding like terms with exponents:** When the bases and exponents are the same, you can add the coefficients. 3. **Rewrite the expression:** $$2^{k+1} + 2^{k+1} = 1 \cdot 2^{k+1} + 1 \cdot 2^{k+1}$$ 4. **Add the coefficients:** $$= (1 + 1) \cdot 2^{k+1} = 2 \cdot 2^{k+1}$$ 5. **Use the property of exponents:** $$2 \cdot 2^{k+1} = 2^{1} \cdot 2^{k+1} = 2^{1 + k + 1} = 2^{k+2}$$ 6. **Final answer:** $$2^{k+1} + 2^{k+1} = 2^{k+2}$$