Convert the following base-ten numbers to a numerical in the indicated bases:

a. 303 in base three

b. 1876 in base twelve

Sure! To convert base-ten numbers to a different base, you need to understand the concept of place value. Each digit in a number represents a power of the base. Let's go through each conversion step by step:

a. Converting 303 to base three:
To convert 303 to base three, we need to find the appropriate digits for each power of three.

First, divide 303 by the largest power of three that is less than or equal to the number. In this case, it is 3 raised to the power of 4, which is 81. The result is 3 with a remainder of 40.

Next, divide the remainder, 40, by the next smallest power of three, which is 3 raised to the power of 3, equal to 27. The quotient is 1 with a remainder of 13.

Continue this process by dividing the remainder in each step by the next smaller power of three until we reach the smallest power of three, which is 3 raised to the power of 0, equal to 1.

The final quotient will be the digit in base three for each corresponding power of three.

Therefore, in base three, 303 is represented as 11010.

b. Converting 1876 to base twelve:
To convert 1876 to base twelve, we follow a similar process as before.

First, divide 1876 by the largest power of twelve that is less than or equal to the number. In this case, it is 12 raised to the power of 3, which is 1728. The result is 1 with a remainder of 148.

Next, divide the remainder, 148, by the next smallest power of twelve, which is 12 raised to the power of 2, equal to 144. The quotient is 1 with a remainder of 4.

Continue this process by dividing the remainder in each step by the next smaller power of twelve until we reach the smallest power of twelve, which is 12 raised to the power of 0, equal to 1.

The final quotient will be the digit in base twelve for each corresponding power of twelve.

Therefore, in base twelve, 1876 is represented as 1144.