To find the equivalent in base 10 of the first 4-digit number in base 3, we start by identifying the first 4-digit number in base 3.

In base 3, the smallest 4-digit number is $1000_3$. Now, we'll convert this to base 10.

The number $1000_3$ can be expanded as follows: $1000_3 = 1 \cdot 3^3 + 0 \cdot 3^2 + 0 \cdot 3^1 + 0 \cdot 3^0$ Calculating each term: $1 \cdot 3^3 = 1 \cdot 27 = 27$ $0 \cdot 3^2 = 0 \cdot 9 = 0$ $0 \cdot 3^1 = 0 \cdot 3 = 0$ $0 \cdot 3^0 = 0 \cdot 1 = 0$

Adding these together: $27 + 0 + 0 + 0 = 27$

So, the equivalent in base 10 of the first 4-digit number in base 3 ($1000_3$) is $27_{10}$.

Would you like more details or have any questions? Here are some related questions to explore:

1. What is the base 10 equivalent of the 4-digit number $2001_3$?
2. How do you convert any base 3 number to base 10?
3. What is the largest 4-digit number in base 3, and what is its base 10 equivalent?
4. How do base conversions between non-decimal systems work in general?
5. What is the first 5-digit number in base 3, and its base 10 equivalent?
6. How does base 3 compare to binary in terms of digit length for the same number?
7. Can you find the base 3 equivalent of the number $81_{10}$?
8. What is the smallest 3-digit number in base 3, and its base 10 equivalent?

Tip: Always start conversions from the highest place value when working with non-decimal bases to avoid confusion.