To convert $-232330_6$ (base 6) to base 4, we will follow these steps:

Step 1: Convert from base 6 to base 10

The number $-232330_6$ in base 6 is expanded as follows:

$-232330_6 = -(2 \times 6^5 + 3 \times 6^4 + 2 \times 6^3 + 3 \times 6^2 + 3 \times 6^1 + 0 \times 6^0)$

Now, let's calculate each term:

$2 \times 6^5 = 2 \times 7776 = 15552$ $3 \times 6^4 = 3 \times 1296 = 3888$ $2 \times 6^3 = 2 \times 216 = 432$ $3 \times 6^2 = 3 \times 36 = 108$ $3 \times 6^1 = 3 \times 6 = 18$ $0 \times 6^0 = 0$

Now, sum them up:

$15552 + 3888 + 432 + 108 + 18 = 20000$

Thus, $-232330_6 = -20000_{10}$.

Step 2: Convert from base 10 to base 4

Next, we need to convert $-20000_{10}$ to base 4. We repeatedly divide by 4 and record the remainders:

$20000 \div 4 = 5000 \ \text{remainder} \ 0$ $5000 \div 4 = 1250 \ \text{remainder} \ 0$ $1250 \div 4 = 312 \ \text{remainder} \ 2$ $312 \div 4 = 78 \ \text{remainder} \ 0$ $78 \div 4 = 19 \ \text{remainder} \ 2$ $19 \div 4 = 4 \ \text{remainder} \ 3$ $4 \div 4 = 1 \ \text{remainder} \ 0$ $1 \div 4 = 0 \ \text{remainder} \ 1$

Reading the remainders from bottom to top, $-20000_{10}$ in base 4 is $103203200_4$.

Final Answer:

$-232330_6 = -103203200_4$

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. Convert $234520_6$ to base 4.
2. Convert $-45321_6$ to base 8.
3. Convert $101101_2$ to base 6.
4. Convert $321032_4$ to base 6.
5. Convert $-32110_6$ to base 5.

Tip: When converting between bases, it is often easier to first convert to base 10, then to the desired base.