The base three number $12012_3$ is equal to which base ten number?
$12012_3=1\cdot3^4+2\cdot3^3+0\cdot3^2+1\cdot3^1+2\cdot3^0=81+54+3+2=\boxed{140}$.
140
Level 1
Number Theory
Problem: The base three number $12012_3$ is equal to which base ten number?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To convert the base three number $12012_3$ to base ten, we need to understand the positional value of each digit in base three. Each digit represents a power of three, starting from the rightmost digit, which is the least significant.

The number $12012_3$ can be broken down as follows:
- The rightmost digit (2) is in the $3^0$ place.
- The next digit (1) is in the $3^1$ place.
- The next digit (0) is in the $3^2$ place.
- The next digit (2) is in the $3^3$ place.
- The leftmost digit (1) is in the $3^4$ place.

Now, we calculate the value of each digit multiplied by its corresponding power of three:
- The leftmost digit: $1 \times 3^4 = 1 \times 81 = 81$.
- The next digit: $2 \times 3^3 = 2 \times 27 = 54$.
- The next digit: $0 \times 3^2 = 0 \times 9 = 0$.
- The next digit: $1 \times 3^1 = 1 \times 3 = 3$.
- The rightmost digit: $2 \times 3^0 = 2 \times 1 = 2$.

Next, we sum these values to find the total in base ten:
$81 + 54 + 0 + 3 + 2 = 140$.

Thus, the base three number $12012_3$ is equal to the base ten number $140$.<|END_OF_LATENT|> The final answer is $140$.