To solve the expressions involving the numbers with bases in the form of __--21, __--24, __--28, __--42, and __--52, we need to follow these steps:
- Convert the numbers from their respective bases to decimal.
- Perform the arithmetic operations.
- Convert the result back to the required base if necessary.
Let's break this down.
Step 1: Base Conversion
Let's assume the numbers are written as follows:
- 3_--21: This is a number in base 21.
- 7_--21: This is another number in base 21.
In base b, the number 3_b represents \(3 \times b^0 = 3\) in decimal.
Similarly, for 7_b, it represents \(7 \times b^0 = 7\).
So:
- \(3_{21} = 3\)
- \(7_{21} = 7\)
Now we perform addition of these two numbers:
\[ 3 + 7 = 10 \]
Now we recognize that \(10_{21}\) still means the same \(10\) when viewed, but the number is the same in decimal.
Step 2: Second Part of the Expression
Now let's evaluate:
\(_--24 + _--28\)
Here we don’t have specific numbers to work with in those bases. However, from the previous step, we assume they are summations awaiting specific base interpretations.
Suppose the results represent a range of numbers in other bases not fully specified: Yet we can structure this numeral placement.
Step 3: Evaluating Other Expressions
For the numbers:
-
\(10_{21} = 10_{21}\) remains 10.
-
\(10_{42}\): This too means \(10_{42}=10\).
Lastly for \(_--52\), this notation isn't specific, yet retains the general number scheme.
Conclusion:
Since we lack distinct numbers for parts of the second equation and thus cannot render a direct solution for it, we can resolve the equations we clearly interpret:
- Therefore, \(3_--21 + 7_--21 = 10_{21}\)
- And \(10_{42}\) = 10.
The _--52 remains unspecified for proper numeric context but might lead to characteristic values in numeral specifications based on exact computation needed for clarity.
If you can provide additional information or a specific number in the formats indicated, I can offer a more precise resolution!