Convert 110 base 3 to 8

Answer 1

To convert the number \(110\) from base \(3\) to base \(8\), we will first convert it to base \(10\) and then from base \(10\) to base \(8\).

Step 1: Convert from base \(3\) to base \(10\)

The base \(3\) number \(110\) can be expanded as follows:

\[ 1 \times 3^2 + 1 \times 3^1 + 0 \times 3^0 \]

Calculating each term:

• \(1 \times 3^2 = 1 \times 9 = 9\)
• \(1 \times 3^1 = 1 \times 3 = 3\)
• \(0 \times 3^0 = 0 \times 1 = 0\)

Now, adding these values together:

\[ 9 + 3 + 0 = 12 \]

So, \(110_3\) is equal to \(12_{10}\).

Step 2: Convert from base \(10\) to base \(8\)

Now, we convert \(12_{10}\) to base \(8\).

To do this, we divide the number by \(8\) and keep track of the remainders:

1. \(12 \div 8 = 1\) with a remainder of \(4\)
2. \(1 \div 8 = 0\) with a remainder of \(1\)

Now, we read the remainders from bottom to top, which gives us \(14\) in base \(8\).

Thus, \(12_{10}\) is equal to \(14_8\).

Conclusion

Therefore, the base \(3\) number \(110\) is equal to \(14\) in base \(8\):

\[ 110_3 = 14_8 \]