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Sagot :
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]
### 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]
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