I saw the solution video of the following problem on Topic Based Quant Quiz on Exponents. I still don’t understand why we added 54, 9, 6 and 1 at the end to get the answer.
Problem -
The decimal system comprises ten digits, 00 through 99, and utilizes powers of 1010 to represent numbers. For example, the number 271271 can be rewritten as the following: 2×102+7×101+1×1002×102+7×101+1×100.
In the same vein, the ternary number system comprises three digits, 00, 11, and 22, and utilizes powers of 33 to represent numbers. What does the ternary number 21212121 equate to in the decimal system?
The decimal system comprises ten digits, 0 to 9, and utilizes powers of 10 to represent numbers. For example, 271 = 2 x 10^2 + 7 x 10^1 + 1 x 10^0.
In the same vein, the ternary number system uses three digits 0, 1, and 2 and uses powers of 3 to represent numbers. What does the ternary number 2121 equal in decimal?
For this question, 2121 = 2 x 3^3 + 1 x 3^2 + 2 x 3^1 + 1 x 3^0 = 2 x 27 + 9 + 6 + 1 = 54 + 9 + 6 + 1 = 70. Each digit tells us how many of the corresponding power of 3 we have.
If 2121 were already in decimal, then it equals 2 x 10^3 + 1 x 10^2 + 2 x 10^1 + 1 x 10^0. Each digit tells us how many of the corresponding power of ten we have.
0101 in binary (powers of 2) is 0 x 2^3 +1 x 2^2 + 0 x 2^1 + 1 x 2^0 = 4 + 1 = 5. Just as an additional example of converting numbers from other number systems to decimal.