To convert the base-7 number $232055_7$ into a base-10 number, follow these steps:

We expand the base-7 number using powers of 7:

$232055_7 = 2 \times 7^5 + 3 \times 7^4 + 2 \times 7^3 + 0 \times 7^2 + 5 \times 7^1 + 5 \times 7^0$

Now, calculate each term:

$2 \times 7^5 = 2 \times 16807 = 33614$ $3 \times 7^4 = 3 \times 2401 = 7203$ $2 \times 7^3 = 2 \times 343 = 686$ $0 \times 7^2 = 0$ $5 \times 7^1 = 5 \times 7 = 35$ $5 \times 7^0 = 5 \times 1 = 5$

Now sum them all up:

$33614 + 7203 + 686 + 0 + 35 + 5 = 41543$

So, the base-10 equivalent of $232055_7$ is 41543.

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. Convert $123456_7$ to base-10.
2. Convert $54321_7$ to base-10.
3. What is the base-7 representation of $1000_{10}$?
4. How do you convert a number from base-10 to base-7 manually?
5. How can you verify your base conversion result?

Tip: Always double-check each term in the expansion to avoid calculation errors during base conversions.