Express 413⁷ in base 5

The digits in base 7 represent powers of 7.

$$413_7 = 4 \times 7^2 + 1 \times 7^1 + 3 \times 7^0$$ $$413_7 = 4 \times 49 + 1 \times 7 + 3 \times 1$$ $$413_7 = 196 + 7 + 3$$ $$413_7 = 206_{10}$$

We use division with remainder by 5.

$$206 \div 5 = 41 \text{ remainder } 1$$ $$41 \div 5 = 8 \text{ remainder } 1$$ $$8 \div 5 = 1 \text{ remainder } 3$$ $$1 \div 5 = 0 \text{ remainder } 1$$

Reading the remainders from bottom to top, we get 1311.

So, $413_7 = 206_{10} = 1311_5$.

The final answer is $1311_5$.