To solve the problem of adding the two numbers [tex]\(221_{\text{six}}\)[/tex] and [tex]\(45_{\text{six}}\)[/tex] and obtaining the result in base-6, we will break it down into several steps.
### Step 1: Convert [tex]\(221_{\text{six}}\)[/tex] to Decimal (Base-10)
The number [tex]\(221_{\text{six}}\)[/tex] is in base-6. To convert it to decimal:
[tex]\[
221_{\text{six}} = 2 \times 6^2 + 2 \times 6^1 + 1 \times 6^0
\][/tex]
[tex]\[
= 2 \times 36 + 2 \times 6 + 1 \times 1
\][/tex]
[tex]\[
= 72 + 12 + 1
\][/tex]
[tex]\[
= 85_{\text{ten}}
\][/tex]
### Step 2: Convert [tex]\(45_{\text{six}}\)[/tex] to Decimal (Base-10)
The number [tex]\(45_{\text{six}}\)[/tex] is also in base-6. To convert it to decimal:
[tex]\[
45_{\text{six}} = 4 \times 6^1 + 5 \times 6^0
\][/tex]
[tex]\[
= 4 \times 6 + 5 \times 1
\][/tex]
[tex]\[
= 24 + 5
\][/tex]
[tex]\[
= 29_{\text{ten}}
\][/tex]
### Step 3: Add the Decimal Numbers
Now, we add the two numbers we have converted to decimal:
[tex]\[
85_{\text{ten}} + 29_{\text{ten}} = 114_{\text{ten}}
\][/tex]
### Step 4: Convert Decimal Sum Back to Base-6
To convert [tex]\(114_{\text{ten}}\)[/tex] back to base-6, we repeatedly divide by 6 and record the remainders:
[tex]\[
114 \div 6 = 19 \text{ remainder } 0
\][/tex]
[tex]\[
19 \div 6 = 3 \text{ remainder } 1
\][/tex]
[tex]\[
3 \div 6 = 0 \text{ remainder } 3
\][/tex]
Reading the remainders from bottom to top, [tex]\(114_{\text{ten}}\)[/tex] converts to:
[tex]\[
310_{\text{six}}
\][/tex]
### Conclusion
Thus, the sum of [tex]\(221_{\text{six}}\)[/tex] and [tex]\(45_{\text{six}}\)[/tex] is:
[tex]\[
\boxed{310_{\text{six}}}
\][/tex]
