1. **State the problem:** Simplify $$4 + i^5$$ and $$2i^{13} - (i^{41} + i)$$ where $$i$$ is the imaginary unit with property $$i^2 = -1$$. 2. **Recall powers of $$i$$:** $$i^1 = i$$ $$i^2 = -1$$ $$i^3 = -i$$ $$i^4 = 1$$ Powers of $$i$$ repeat every 4 steps. 3. **Simplify $$i^5$$:** $$i^5 = i^{4+1} = i^4 \cdot i^1 = 1 \cdot i = i$$ So, $$4 + i^5 = 4 + i$$. 4. **Simplify $$i^{13}$$:** Find remainder of $$13$$ divided by $$4$$: $$13 \mod 4 = 1$$ Thus, $$i^{13} = i^{4 \cdot 3 + 1} = (i^4)^3 \cdot i^1 = 1 \cdot i = i$$ 5. **Simplify $$i^{41}$$:** Find remainder of $$41$$ divided by $$4$$: $$41 \mod 4 = 1$$ So, $$i^{41} = (i^4)^{10} \cdot i^1 = 1 \cdot i = i$$ 6. **Simplify the expression:** $$2i^{13} - (i^{41} + i) = 2i - (i + i) = 2i - 2i = 0$$. **Final answers:** $$4 + i^5 = 4 + i$$ $$2i^{13} - (i^{41} + i) = 0$$