To convert the number \(221\) in base \(3\) to base \(8\), we first need to convert it from base \(3\) to decimal (base \(10\)), and then from decimal to base \(8\).
Step 1: Convert from base 3 to decimal
The number \(221\) in base \(3\) can be broken down as follows:
\[ 2 \times 3^2 + 2 \times 3^1 + 1 \times 3^0 \]
Calculating each term:
- \(2 \times 3^2 = 2 \times 9 = 18\)
- \(2 \times 3^1 = 2 \times 3 = 6\)
- \(1 \times 3^0 = 1 \times 1 = 1\)
Now, add these values together:
\[ 18 + 6 + 1 = 25 \]
So, the decimal (base \(10\)) equivalent of \(221\) in base \(3\) is \(25\).
Step 2: Convert from decimal to base 8
Now, we need to convert \(25\) from decimal to base \(8\). We do this by repeatedly dividing the number by \(8\) and recording the remainders.
- \(25 \div 8 = 3\) remainder \(1\)
- \(3 \div 8 = 0\) remainder \(3\)
Now, write the remainders in reverse order to get the base \(8\) number:
Starting from the last remainder we obtained (which is the most significant digit):
- Remainders are \(3\) and \(1\)
Thus, \(25\) in base \(8\) is \(31\).
Final Result
The number \(221\) in base \(3\) is \(31\) in base \(8\).
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