To convert the base 7 number $5624_7$ to base 10, follow these steps:

Each digit is multiplied by 7 raised to the power of its position (counting from right to left, starting at 0).

$5624_7 = 5 \cdot 7^3 + 6 \cdot 7^2 + 2 \cdot 7^1 + 4 \cdot 7^0$

Now calculate each term:

$5 \cdot 7^3 = 5 \cdot 343 = 1715$ $6 \cdot 7^2 = 6 \cdot 49 = 294$ $2 \cdot 7^1 = 2 \cdot 7 = 14$ $4 \cdot 7^0 = 4 \cdot 1 = 4$

Now, add these values together:

$1715 + 294 + 14 + 4 = 2027$

So, the base 10 equivalent of $5624_7$ is $2027$.

Would you like any details on this conversion?

Here are five related questions to explore:

1. How would you convert a base 10 number to base 7?
2. What is the base 7 equivalent of the base 10 number 2027?
3. How do you convert base 5 numbers to base 10?
4. What are common strategies for converting between different bases?
5. How do powers of 7 change as you move to higher positions in a number?

Tip: When converting from any base to base 10, always multiply each digit by the base raised to the power of its positional index.