Sure, let's convert each of these base-six numbers to base-ten numbers step by step.
### 1. Converting [tex]\(555_{\text{six}}\)[/tex] to base ten:
The notation [tex]\(555_{\text{six}}\)[/tex] means that each digit represents a power of six.
[tex]\[
555_{\text{six}} = 5 \cdot 6^2 + 5 \cdot 6^1 + 5 \cdot 6^0
\][/tex]
Breaking it down:
[tex]\[
5 \cdot 6^2 = 5 \cdot 36 = 180
\][/tex]
[tex]\[
5 \cdot 6^1 = 5 \cdot 6 = 30
\][/tex]
[tex]\[
5 \cdot 6^0 = 5 \cdot 1 = 5
\][/tex]
Adding these together:
[tex]\[
180 + 30 + 5 = 215
\][/tex]
So, [tex]\(555_{\text{six}} = 215_{\text{ten}}\)[/tex].
### 2. Converting [tex]\(215_{\text{six}}\)[/tex] to base ten:
[tex]\[
215_{\text{six}} = 2 \cdot 6^2 + 1 \cdot 6^1 + 5 \cdot 6^0
\][/tex]
Breaking it down:
[tex]\[
2 \cdot 6^2 = 2 \cdot 36 = 72
\][/tex]
[tex]\[
1 \cdot 6^1 = 1 \cdot 6 = 6
\][/tex]
[tex]\[
5 \cdot 6^0 = 5 \cdot 1 = 5
\][/tex]
Adding these together:
[tex]\[
72 + 6 + 5 = 83
\][/tex]
So, [tex]\(215_{\text{six}} = 83_{\text{ten}}\)[/tex].
### 3. Converting [tex]\(110_{\text{six}}\)[/tex] to base ten:
[tex]\[
110_{\text{six}} = 1 \cdot 6^2 + 1 \cdot 6^1 + 0 \cdot 6^0
\][/tex]
Breaking it down:
[tex]\[
1 \cdot 6^2 = 1 \cdot 36 = 36
\][/tex]
[tex]\[
1 \cdot 6^1 = 1 \cdot 6 = 6
\][/tex]
[tex]\[
0 \cdot 6^0 = 0 \cdot 1 = 0
\][/tex]
Adding these together:
[tex]\[
36 + 6 + 0 = 42
\][/tex]
So, [tex]\(110_{\text{six}} = 42_{\text{ten}}\)[/tex].
### 4. Converting [tex]\(1000_{\text{six}}\)[/tex] to base ten:
[tex]\[
1000_{\text{six}} = 1 \cdot 6^3 + 0 \cdot 6^2 + 0 \cdot 6^1 + 0 \cdot 6^0
\][/tex]
Breaking it down:
[tex]\[
1 \cdot 6^3 = 1 \cdot 216 = 216
\][/tex]
[tex]\[
0 \cdot 6^2 = 0 \cdot 36 = 0
\][/tex]
[tex]\[
0 \cdot 6^1 = 0 \cdot 6 = 0
\][/tex]
[tex]\[
0 \cdot 6^0 = 0 \cdot 1 = 0
\][/tex]
Adding these together:
[tex]\[
216 + 0 + 0 + 0 = 216
\][/tex]
So, [tex]\(1000_{\text{six}} = 216_{\text{ten}}\)[/tex].
### 5. Converting [tex]\(82_{\text{six}}\)[/tex] to base ten:
[tex]\[
82_{\text{six}} = 8 \cdot 6^1 + 2 \cdot 6^0
\][/tex]
Breaking it down:
[tex]\[
8 \cdot 6^1 = 8 \cdot 6 = 48
\][/tex]
[tex]\[
2 \cdot 6^0 = 2 \cdot 1 = 2
\][/tex]
Adding these together:
[tex]\[
48 + 2 = 50
\][/tex]
So, [tex]\(82_{\text{six}} = 50_{\text{ten}}\)[/tex].
### Summary:
[tex]\[
\begin{align*}
555_{\text{six}} & = 215_{\text{ten}} \\
215_{\text{six}} & = 83_{\text{ten}} \\
110_{\text{six}} & = 42_{\text{ten}} \\
1000_{\text{six}} & = 216_{\text{ten}} \\
82_{\text{six}} & = 50_{\text{ten}}
\end{align*}
\][/tex]
