Answer:
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Step-by-step explanation:
To solve this, let's first convert each number from base 5 to base 10, perform the operations, and then convert the result back to base 5.
1. Convert 204 base 5 to base 10:
\[204_{5} = 2 \times 5^2 + 0 \times 5^1 + 4 \times 5^0 = 2 \times 25 + 0 \times 5 + 4 \times 1 = 50 + 0 + 4 = 54_{10}\]
2. Convert 243 base 5 to base 10:
\[243_{5} = 2 \times 5^2 + 4 \times 5^1 + 3 \times 5^0 = 2 \times 25 + 4 \times 5 + 3 \times 1 = 50 + 20 + 3 = 73_{10}\]
3. Convert -21 base 5 to base 10:
The negative sign is represented by subtracting the value in base 5. So, -21 base 5 is:
\[-21_{5} = -(2 \times 5^1 + 1 \times 5^0) = -(2 \times 5 + 1 \times 1) = -(10 + 1) = -11_{10}\]
Now, let's perform the operations in base 10:
\[54_{10} + 73_{10} - 11_{10} = 54 + 73 - 11 = 116_{10}\]
Now, we need to convert 116 base 10 back to base 5:
\[116_{10} = 4 \times 5^2 + 3 \times 5^1 + 1 \times 5^0 = 4 \times 25 + 3 \times 5 + 1 \times 1 = 100 + 15 + 1 = 114_{5}\]
So, \(204_{5} + 243_{5} - 21_{5} = 114_{5}\).
