To convert the binary number [tex]\( (111100111110001)_2 \)[/tex] to its octal equivalent, follow these steps:
### Step 1: Convert the binary number to its decimal equivalent.
First, break down the binary number [tex]\( 111100111110001 \)[/tex]. Each digit represents a power of 2, starting from the rightmost digit (which is [tex]\( 2^0 \)[/tex]). Here's how the conversion to a decimal number works:
[tex]\[
\begin{align*}
1 \times 2^{14} & = 16384 \\
1 \times 2^{13} & = 8192 \\
1 \times 2^{12} & = 4096 \\
1 \times 2^{11} & = 2048 \\
0 \times 2^{10} & = 0 \\
0 \times 2^{9} & = 0 \\
1 \times 2^{8} & = 256 \\
1 \times 2^{7} & = 128 \\
1 \times 2^{6} & = 64 \\
1 \times 2^{5} & = 32 \\
0 \times 2^{4} & = 0 \\
0 \times 2^{3} & = 0 \\
0 \times 2^{2} & = 0 \\
0 \times 2^{1} & = 0 \\
1 \times 2^{0} & = 1 \\
\end{align*}
\][/tex]
Now, sum up these values to obtain the decimal equivalent:
[tex]\[
16384 + 8192 + 4096 + 2048 + 256 + 128 + 64 + 32 + 1 = 31217
\][/tex]
So, the decimal equivalent of the binary number [tex]\( 111100111110001 \)[/tex] is [tex]\( 31217 \)[/tex].
### Step 2: Convert the decimal number to its octal equivalent.
Next, we convert the decimal number [tex]\( 31217 \)[/tex] to an octal number by dividing by 8 and keeping track of the remainders.
[tex]\[
\begin{align*}
31217 \div 8 & = 3902 \quad \text{remainder} \ 1 \\
3902 \div 8 & = 487 \quad \text{remainder} \ 6 \\
487 \div 8 & = 60 \quad \text{remainder} \ 7 \\
60 \div 8 & = 7 \quad \text{remainder} \ 4 \\
7 \div 8 & = 0 \quad \text{remainder} \ 7 \\
\end{align*}
\][/tex]
Write the remainders in the reverse order of their computation:
[tex]\[
74761_8
\][/tex]
So, the octal equivalent of the binary number [tex]\( 111100111110001_2 \)[/tex] is [tex]\( 74761_8 \)[/tex].
Therefore, [tex]\((111100111110001)_2 = (74761)_8\)[/tex].
