Convert `(332.22)_(4)` to base 5.

Options

A. `(222.bar(30))_(5)`
B. `(222.30)_(5)`
C. `(222.33)_(5)`
D. None of these

Answer

To convert the number \( (332.22)_4 \) to base 5, we will follow these steps:

### Step 1: Convert from Base 4 to Decimal
First, we need to convert the base 4 number \( (332.22)_4 \) to decimal (base 10).

The number can be broken down as follows:
\[ (332.22)_4 = 3 \times 4^2 + 3 \times 4^1 + 2 \times 4^0 + 2 \times 4^{-1} + 2 \times 4^{-2} \]

Calculating each term:
- \( 3 \times 4^2 = 3 \times 16 = 48 \)
- \( 3 \times 4^1 = 3 \times 4 = 12 \)
- \( 2 \times 4^0 = 2 \times 1 = 2 \)
- \( 2 \times 4^{-1} = 2 \times \frac{1}{4} = 0.5 \)
- \( 2 \times 4^{-2} = 2 \times \frac{1}{16} = 0.125 \)

Now, summing these values:
\[ 48 + 12 + 2 + 0.5 + 0.125 = 62.625 \]

So, \( (332.22)_4 = 62.625_{10} \).

### Step 2: Convert from Decimal to Base 5
Next, we will convert \( 62.625_{10} \) to base 5.

#### Convert the integer part (62) to base 5:
1. Divide 62 by 5:
- \( 62 \div 5 = 12 \) remainder \( 2 \)
2. Divide 12 by 5:
- \( 12 \div 5 = 2 \) remainder \( 2 \)
3. Divide 2 by 5:
- \( 2 \div 5 = 0 \) remainder \( 2 \)

Reading the remainders from bottom to top, we get \( (222)_5 \).

#### Convert the fractional part (0.625) to base 5:
1. Multiply 0.625 by 5:
- \( 0.625 \times 5 = 3.125 \) (integer part is 3)
2. Take the fractional part (0.125) and multiply by 5:
- \( 0.125 \times 5 = 0.625 \) (integer part is 0)
3. Take the fractional part (0.625) and multiply by 5 again:
- \( 0.625 \times 5 = 3.125 \) (integer part is 3)

Thus, the fractional part in base 5 is \( (0.303)_5 \).

### Final Result
Combining both parts, we get:
\[ (332.22)_4 = (222.303)_5 \]

### Conclusion
The answer is \( (222.30)_5 \) (the repeating part can be noted as \( 0.30 \) repeating).